Nilpotency, almost nonnegative curvature and \\ the gradient push
Vitali Kapovitch
We show that almost nonnegatively curved $m$-manifolds are, up to finite cover, nilpotent spaces in the sense of homotopy theory and have $C(m)$-nilpotent fundamental groups. We also show that up to a finite cover almost nonnegatively curved manifolds are fiber bundles with simply connected fibers over nilmanifolds.
math0506273.tex
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\begin{document}
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\title[Nilpotency, almost nonnegative curvature and the gradient push]
{Nilpotency, almost nonnegative curvature and \\ the gradient push}


%\title[Nilpotency and almost nonnegative curvature]
%{Nilpotency and almost nonnegative curvature}

\thanks{\it 2000 AMS Mathematics Subject Classification:\rm\
53C20. Keywords: nonnegative curvature, nilpotent}\rm
\thanks{\it The first two authors were supported in part by NSF grants. 
%\# DMS-0204187. 
The last author's research was supported by a 
DFG Heisenberg Fellowship.}
\author{Vitali Kapovitch}
\address{Vitali Kapovitch\\Department of Mathematics\\University of M\"unster\\
48149 M\"unster, Germany}\email{kapowitc@math.uni-muenster.de}
\author{ Anton Petrunin }\address{Anton Petrunin\\ Department of Mathematics\\ Pennsylvania State University\\
University Park, State College, PA 16802
}\email{petrunin@math.psu.edu}
\author{Wilderich Tuschmann}\address{Wilderich Tuschmann
\\Sonderforschungsbereich ``Geometrische Strukturen in der Mathematik''
\\Westf\"alische Wilhelms-Universit\"at M\"unster\\
D-48149 M\"unster, Germany}\email{wtusch@math.uni-muenster.de}



\maketitle
\begin{abstract}
We show that almost nonnegatively curved $m$-manifolds are, up to finite cover,
nilpotent spaces in the sense of homotopy theory
and have $C(m)$-nilpotent fundamental groups.
We also show that up to a finite cover 
almost nonnegatively curved manifolds are fiber bundles 
with simply connected fibers over nilmanifolds.
\end{abstract}


\section{Introduction}

Almost nonnegatively curved manifolds were introduced by Gromov 
in the late 70s ~\cite{G4},
with the most significant contributions to their study made by
Yamaguchi in \cite{Yam} and Fukaya and Yamaguchi in ~\cite{FY}.
Building on their ideas,
in the present article 
we establish several 
new properties of these %almost nonnegatively curved
manifolds which yield, in particular, 
new topological obstructions to almost nonnegative curvature.


A closed smooth manifold is  said 
to be almost nonnegatively curved
if
it can Gromov-Hausdorff converge to a single point
%it can collapse to a  point in Gromov-Hausdorff topology 
under a lower curvature bound.  
By rescaling, this definition is equivalent to the following one,
which we will employ throughout this article.
\begin{defn}
A closed smooth manifold $M$ is called almost nonnegatively curved 
if it admits a sequence of Riemannian metrics $\{g_n\}_{n\in\NN}$ 
whose sectional curvatures and diameters satisfy
$\sec(M,g_n)\ge -1/n$ and $\diam(M,g_n)\le 1/n$.
\end{defn}

Almost nonnegatively curved manifolds 
generalize almost flat as well as nonnegatively 
curved manifolds. 
One main source of examples 
comes from a theorem of Fukaya and Yamaguchi. 
It states that
if $F\to E\to B$ is a fiber bundle 
over an almost nonnegatively curved manifold $B$
whose fiber $F$ is compact 
and admits a nonnegatively curved metric %$g$ 
which is invariant under the
structure group, %of the bundle 
%acts on $(F,g)$ by isometries, 
then the total space $E$ is almost nonnegatively curved~\cite{FY}.
Further examples are given by closed manifolds 
which admit a cohomogeneity one action of 
a compact Lie group %are almost nonnegatively curved 
(compare~\cite{ST}).


In this work we combine collapsing techniques 
with a non-smooth analogue of the gradient flow of  
concave functions which we call the ``gradient push''. 
This notion is based  on the construction  of gradient curves 
of $\lambda$-concave 
functions used in ~\cite{PP} 
and bears many similarities to the Sharafutdinov retraction~\cite{Shar}.
The gradient push plays a key role in the proofs 
of two of the three main results in this paper, 
and we  believe that it should also prove useful 
for dealing with other problems related to
collapsing under a lower curvature bound.

\subsection{} To put the main theorems of the present work into perspective, 
let us first briefly recall some  previously known results:

%\pagebreak 
Let $M=M^m$ be an almost nonnegatively curved $m$-manifold. 
\begin{enumerate}[$*$]
\item Gromov proved in ~\cite{G1}
that the minimal number of generators of the fundamental
group $\pi_1(M)$ of $M$ can be estimated by a constant $C_1(m)$
depending only on $m$,
and in ~\cite{G2} that the sum of Betti numbers of $M$ with 
respect to any field of coefficients
does not exceed some uniform constant $C_2=C_2(m)$.
\item
Yamaguchi showed that, up to a finite cover, $M$
fibers over a flat $b_1(M;\R)$-dimensionsal torus
and that $M^m$ is diffeomorphic to a torus if $b_1(M;\R)=m$~\cite{Yam}.
\item
Fukaya and Yamaguchi proved that $\pi_1(M)$ is almost nilpotent,
i.e., contains a nilpotent subgroup of finite index,
and also that  $\pi_1(M)$  is $C_3(m)$-solvable, i.e, 
contains a solvable subgroup of index at most $C_3(m)$~\cite{FY}.
\item If a closed manifold has  negative Yamabe constant, 
then it cannot volume collapse with scalar curvature bounded from below 
(see \cite{Sch,Lb}).  In particular, no such manifold 
can be almost nonnegatively curved.
\item The $\hat{A}$-genus of a closed spin manifold $X$ 
of almost nonnegative Ricci curvature 
satisfies the inequality $\hat{A}(X)\le 2^{\dim(X)/2}$ 
(\cite{Ga}, \cite[page 41]{G3}). 
\end{enumerate}




Let us now state  the main results of this article. 

\subsection{}
Our first  result concerns the hitherto unexplored relation
between  curvature bounds and the actions of the fundamental group
on the higher homotopy groups.



Recall that an action by automorphisms of a group $G$ on an abelian group $V$ 
is called nilpotent if $V$ 
admits a finite sequence of $G$-invariant subgroups
 $$V=V_0\supset V_1\supset\ldots\supset V_k=0$$
such that the induced action of $G$ on $V_i/V_{i+1}$ is trivial for any $i$.
A connected CW-complex $X$ is called {\it nilpotent}
if $\pi_1(X)$ is a nilpotent group that operates nilpotently
on $\pi_k(X)$ for every $k\ge 2$. 

Nilpotent spaces play an important role in topology
since they enjoy some of the best 
homotopy-theoretic properties of simply connected spaces,
like a Whitehead theorem or reasonable Postnikov towers.
Furthermore, unlike the category of simply connected spaces,
the category of nilpotent ones is closed under
many constructions such as the based loop space functor
or the formation of function spaces,
and group-theoretic functors, like localization and completion,
have topological extensions in this category.

\begin{main}[Nilpotency Theorem]\label{intro:main1}
Let $M$ be a closed almost nonnegatively curved manifold. 
Then a finite cover of $M$ is a nilpotent space.
\end{main}

It would be interesting to know 
whether  the order of this covering 
can be estimated solely in terms of the dimension of $M$.

\begin{ex} 
Let  $h\co S^3 \times S^3\to S^3 \times S^3$ be defined by $h:(x,y)\mapsto (xy,yxy)$. 
This map  is a diffeomorphism with the inverse given by  $h^{-1}\co (u,v)\mapsto (u^2v^{-1},vu^{-1})$. 
The induced map $h_*$ on $\pi_3( S^3 \times S^3)$ is given by the matrix
$A_h=\left(\begin{matrix}
1&1\\ 1&2
\end{matrix}\right)$. Notice that the eigenvalues of $A_h$ 
are different from $1$ in absolute value.
Let $M$ be the mapping cylinder of $h$.
Clearly, $M$ has the structure of a fiber bundle $S^3 \times S^3 \to M \to S^1$, and the 
action of $\pi_1(M)\cong\ZZ$ on $\pi_3(M)\cong \ZZ^2$
is generated by  $A_h$. 
In particular, $M$ is not a nilpotent space and hence, by Theorem~\ref{intro:main1},
it does not admit almost nonnegative curvature.
This fact doesn't follow from any previously known results.
\end{ex}

\subsection{}
Our next main result provides an affirmative answer to a conjecture of Fukaya and Yamaguchi~\cite[Conjecture 0.15]{FY}.

\begin{main}[$C$-Nilpotency Theorem for $\pi_1$]\label{intro:main2}

Let $M$ be an almost nonnegatively curved $m$-manifold.  
Then $\pi_1(M)$ is $C(m)$-nilpotent,
i.e., 
$\pi_1(M)$ contains a nilpotent subgroup of index at most $C(m)$.

\end{main}
Notice  that Theorem ~\ref{intro:main2} is new even 
for manifolds of nonnegative curvature.

\begin{ex} For any $C>0$ there exist
prime numbers $p>q>C$ and a finite group $G_{pq}$ 
of order $p q$
which is solvable but not nilpotent.
In particular, $G_{pq}$ does not contain any nilpotent subgroup
of index less than or equal to $C$.

Whereas none of the results mentioned so far
excludes %any of the groups 
$G_{pq}$
from being the fundamental group of some 
almost nonnegatively curved $m$-manifold,
Theorem~\ref{intro:main2} shows that for $C>C(m)$ none of the groups $G_{pq}$
can be realized as the fundamental group of
such a manifold.
\end{ex}







\subsection{}
In  ~\cite{FY} Fukaya and Yamaguchi also conjectured
that a finite cover 
of an almost nonnegatively Ricci curved manifold $M$ 
fibers over a nilmanifold with a fiber 
which has nonnegative Ricci curvature and whose fundamental group is finite.
This conjecture
\begin{comment}
conjecture, potentially 
reducing the classification of almost nonnegatively
Ricci curved manifolds to simply-connected ones, 
\end{comment}
was later  refuted  by Anderson ~\cite{An}.


It is, on the other hand,  very natural to consider this conjecture 
in the context of almost nonnegative {\sl sectional} curvature.
In fact, here %one can deduce from 
Yamaguchi's fibration theorem (\cite{Yam})
and the results of \cite{FY} easily imply %the following:
that a finite cover of an almost nonnegatively curved manifold  
is the total space of a Serre fibration over a nilmanifold 
with simply connected fibers. 

From mere topology, it is, however, 
not clear whether this fibration 
can actually  always be made into a genuine fiber bundle.
Our next result shows that 
this is indeed true, 
and that 
for manifolds of almost nonnegative {sectional}  curvature
Fukaya's and Yamaguchi's original conjecture essentially does hold.

 
\begin{main}[Fibration Theorem]\label{intro:main3} 

Let $M$ be an almost nonnegatively curved manifold. 
Then a finite cover $\tilde{M}$ of $M$ 
is the total space of a fiber bundle
 \[
 F\to \tilde{M}\to N
 \]
over a nilmanifold $N$ with a simply-connected fiber $F$.  
Moreover, the fiber $F$  
is  almost nonnegatively curved
in the sense of the following definition.

 \end{main}


\smallskip

 \begin{defn} \label{defn:ann}A closed smooth manifold $M$ 
is called almost nonnegatively curved in the generalized sense
if for some nonnegative integer $k$ 
there exists  a sequence of complete Riemannian metrics $g_n$ on  
$M\times \R^k$
and points $p_n\in M\times \R^k$ 
such that 
\begin{enumerate}
\item the sectional curvatures of the metric balls of radius $n$
around $p_n$ satisfy 
$$\sec(B_n(p_n))\ge -1/n;$$ 
\item for $n\to\infty$
the pointed Riemannian manifolds $((M\times \R^k, g_n), p_n)$ 
converge in the pointed Gromov-Hausdorff distance to $(\R^k,0)$;
\item 
the regular fibres over $0$
are diffeomorphic to $M$ for all large $n$.
\end{enumerate}
\end{defn}

Due to Yamaguchi's fibration theorem \cite{Yam},
manifolds which are almost nonnegatively curved in the generalized sense 
play the same central role in collapsing under a lower curvature bound
as almost flat manifolds do
in the Cheeger-Fukaya-Gromov theory 
of collapsing with bounded curvature  (see \cite{CFG}).

It is not known whether 
all manifolds which are 
almost nonnegatively curved %manifolds 
in the generalized sense
are almost nonnegatively curved.
Clearly, if $k=0$, this definition reduces to the standard one.
Moreover, it is easy to see that all results of the present article,
as well as all results about 
almost nonnegatively curved manifolds mentioned earlier
(except possibly for the ones concerning %restrictions on 
the $\hat A$-genus and Yamabe constant), 
%including the results of the present article, 
hold for manifolds which are almost nonnegatively curved 
in the sense of 
Definition~\ref{defn:ann}.


\smallskip


\subsection{}
Let us now describe the structure of the remaining sections of this article.


In section~\ref{s:gradpush}, after providing some necessary background 
from Alexandrov geometry, 
we introduce the gradient push, which is, roughly speaking, the gradient flow of the square of a distance function. 
It serves as one of 
the main technical tools in the proofs of  theorem~\ref{intro:main1} 
and  theorem~\ref{intro:main2}. 



In section \ref{sec:nilp} we prove Theorem~\ref{intro:main1} by a direct application of the gradient push technique.

In section \ref{sec:c-nilp} we  prove Theorem~\ref{intro:main2}.
The proof is also based on the gradient push, but 
is more involved and employs further technical tools such as
``limit fundamental groups'' of Alexandrov spaces.

In section \ref{sec:fib} we prove Theorem~\ref{intro:main3}. 
This section is completely independent from the rest of the article.
%in particular, it does not depend on section 2.

In section 6 we discuss some further open questions related to our results.
% in particular, describe how the techniques of this paper might  be used  to obtain new information about simply connected manifolds of almost nonnegative curvature.

\smallskip

{\it Acknowledgements.}
We would like to express our thanks to the following people 
for helpful conversations during the preparation of this work: 
I.~Belegradek,
V.~Gorbunov, E.~Formanek, I.~Kapovitch, A.~Lytchak, R.~Matveyev, D.~Robinson, 
D.~Sullivan, B.~Wilking, and Yu.~Zarkhin. 

We are also especially grateful to Yunhui Wu for noticing a mistake in the  proof of Theorem ~\ref{thm:cnilp} in the original version of the paper.

\section{ Alexandrov geometry and the gradient push}\label{s:gradpush}

This section 
provides necessary background in Alexandrov geometry
and introduces the gradient push. 
The results of sections 2.1-2.3 are mostly  repeated from ~\cite{PP} and ~\cite{Ptr2}. 
%The reader could use~
The reader may consult \cite{BGP} for a general reference on Alexandrov spaces.


\subsection{ $\lambda$-concave functions}


\begin{defn}{\it (for a space without boundary)} Let $A$ be an Alexandrov space without boundary. 
A Lipschitz function $f\colon A \rightarrow \RR$ is called
$\lambda$-{\it concave} if for any unit speed minimizing geodesic $\gamma$ 
in $A$, the function
$$f\circ\gamma(t)-\lambda t^2/2$$
is concave.
\end{defn}

If $A$ is an Alexandrov space with boundary, then its double $\tilde A$ is also an Alexandrov space (see \cite[5.2]{Per}). 
Let $\text{p}:\tilde A\to A$ be the canonical map. 
Given a function $f$ on $A$, set $\tilde f=f\circ \text{p}$.

\ 

\begin{defn}{\it (for a space with boundary)}  Let $A$ be an Alexandrov space with boundary. A Lipschitz function $f\colon A \rightarrow \RR$ is called
$\lambda$-{\it concave} if for any unit speed 
minimizing geodesic $\gamma$ in $\tilde A$, the function
$$\tilde f\circ\gamma(t)-\lambda t^2/2 $$
is concave.
\end{defn}


\begin{rmk}Notice that the restriction of a linear function on $\RR^n$ to a ball is not $0$-concave in this sense.
\end{rmk}

\begin{rmk}
In the above definitions, the Lipschitz condition is only technical. 
With some extra work, 
all results of this section can be extended to continuous functions.
\end{rmk}

\subsection{Tangent cone and differential} Given a point $p$ in an Alexandrov space  $A$, we denote by $T_p=T_p(A)$ the tangent cone at $p$.

If $d$ denotes the metric of an Alexandrov space $A$, 
let us denote by $\lambda A$ the space $(A,\lambda d)$.
Let $i_\lambda: \lambda A\to A$ be the canonical map. 
The limit of $(\lambda A,p)$ for $\lambda\to\infty$ is  the tangent cone $T_p$ at $p$ (see \cite[ 7.8.1]{BGP}).

\begin{defn} For any function $f\colon A \rightarrow \RR$ the function $d_pf\colon T_p \rightarrow \RR$ such that 
$$d_pf=\lim_{\lambda\to\infty} \lambda(f\circ i_\lambda-f(p))$$
is called the differential of $f$ at $p$.
\end{defn}

It is easy to see that for a $\lambda$-concave function $f$ 
the differential $d_pf$ is defined everywhere, and that $d_pf$
is a $0$-concave function on the tangent cone $T_p$.

\begin{defn} Given a $\lambda$-concave function $f\colon A \rightarrow \RR$, a point $p\in A$
is called critical point of $f$ if $d_pf\le 0$.
\end{defn}
 

\subsection{Gradient curves}
With a slight abuse of notation we will call elements of the tangent cone $T_p$ the ``tangent vectors'' at $p$.
The origin of $T_p$ plays the role of the zero 
vector and is denoted by $o=o_p$.
For a tangent vector $v$ at $p$ we define its absolute value $|v|$ 
as the distance $|ov|$ in $T_p$.
For two tangent vectors $u$ and $v$ at $p$ we can define 
their ``scalar product'' 
$$\langle u, v\rangle=(|u|^2+|v|^2-|uv|^2)/2=|u|\cdot |v|\cos\alpha,$$ where $\alpha=\angle uov$ in $T_p$.

For two points $p,q\in A$ we define $\log_pq$ to be a tangent vector $v$ 
at $p$ 
such that $|v|=|pq|$ and such that the
direction of $v$ coincides with a direction from $p$ to $q$ 
(if such a direction is not unique, we choose any one of them).
Given a curve $\gamma(t)$ in $A$, we denote by $\gamma^+(t)$ 
the right and by 
$\gamma^-(t)$ the left tangent vectors to  $\gamma(t)$, where, respectively,
$$\gamma^\pm(t)\in T_{\gamma(t)},\ \ \ \gamma^\pm(t)=\lim_{\eps\to +0}\frac{\log_{\gamma(t)}\gamma(t\pm\eps)}{\eps}.$$ 
For a real function $f(t)$, $t\in \RR$, we denote by  $f^+(t)$ its right derivative and by $-f^-(t)$ its left derivative. Note that our sign convention (which is chosen to agree with the notion of right and left derivatives of curves) is not quite standard. For example, 
$$ \ \ \text{if}\ \ f(t)=t \ \ \text{then}\ \ f^+(t)\equiv 1 \ \ \text{and}\ \ f^-(t)\equiv -1.$$


\begin{defn}Given a $\lambda$-concave function $f$ on $A$, 
a vector $g\in T_p(A)$ is called a gradient of $f$ at $p\in A$  
(in short:  $g=\nabla_p f$) if

(i) $d_pf(x)\le \langle g , x\rangle\ \hbox{for any}\ x\in T_p$, and

(ii) $d_pf(g) = \langle g,g \rangle .$
\end{defn}


It is easy to see that any $\lambda$-concave function 
has a uniquely defined gradient vector field. 
Moreover, if $d_pf(x) \le 0$ for all $x\in T_p$,
then $\nabla_p f=o$ 
(here $o$ denotes the origin of the tangent cone $T_p$); otherwise, 
 $$\nabla_p f=d_pf(\xi) \xi$$
where $\xi$ is the (necessarily unique) 
unit vector for which the function $d_pf$ attains its maximum.

Moreover, for any minimizing geodesic $\gamma: [a,b]\to U$ 
parameterized by arclength, the following inequality holds:
\begin{equation}\label{e:star}
\langle \gamma^+(a),\nabla_{\gamma(a)} f\rangle +\langle\gamma^-(b),\nabla_{\gamma(b)} f\rangle
\ge -\lambda(b-a) .
\end{equation}

Indeed,

$$
\langle   \gamma^+(a),  \nabla_{\gamma(a)}  f   \rangle +
\langle  \gamma^-(b),   \nabla_{\gamma(b)} f  \rangle
\ge
d_{\gamma(a)}f(\gamma^+(a))+d_{\gamma(b)}f(\gamma^-(b))
=$$
$$=
(f\circ\gamma)^+|_a+(f\circ\gamma)^-|_b
\ge -\lambda(b-a).
$$

\begin{defn}A curve $\alpha\colon [a,b] \rightarrow A$ is called an $f$-{\it gradient curve} if for any $t\in [a,b]$
$$\alpha^+(t)=\nabla_{\alpha(t)}f.$$
\end{defn}

\begin{prop}\label{prop:gradlip} Given a $\lambda$-concave function $f\colon A \rightarrow \RR$ and a point $p\in A$ there is a unique gradient curve $\alpha\colon [0,\infty) \rightarrow A$ such that $\alpha(0)=p$. 

Moreover, if $\alpha$ and $\beta$ are two $f$-gradient curves, then
$$|\alpha(t_1)\beta(t_1)|\le |\alpha(t_0)\beta(t_0)|\exp(\lambda (t_1-t_0))
\ \ \text{for all}\ \ t_1\ge t_0.$$

\end{prop} 

The gradient curve  can be constructed as a limit of broken geodesics,
made up of short segments with directions close to the gradient. 
The convergence, uniqueness, as well as the last inequality 
in Proposition~\ref{prop:gradlip}
follow from
inequality (\ref{e:star}) above, while Corollary~\ref{cor:gradlim} 
below guarantees that the limit is indeed
a gradient curve, having a unique right tangent vector at each point.

\begin{lem}\label{lem:gradcon}
 Let $A_n \buildrel \text{GH} \over \rightarrow A$ be a sequence of 
Alexandrov spaces with curvature $\ge k$ 
which Gromov-Hausdorff converges to an Alexandrov space $A$.

Let $f_n\to f$, where $f_n:A_n\to \RR$ is a sequence of $\lambda$-concave 
functions converging to $f:A\to \RR$. 

Let $p_n\to p$,  where $p_n\in A_n$ and $p\in A$.

Then 
$$|\nabla_p f|\le \liminf_{n\to \infty} |\nabla_{p_n} f_n|.$$
\end{lem}

\begin{cor}\label{cor:gradlim} Given a $\lambda$-concave function $f$ on $A$ 
and a sequence of points $p_n\in A,\  p_n\to p$, we have 
$$|\nabla_p f|\le \liminf_{n\to \infty} |\nabla_{p_n} f|.$$

\end{cor}

\begin{proof}[Proof of Lemma~\ref{lem:gradcon}] Fix an $\eps>0$ and choose $q$ near $p$ such
that 
$$\frac{f(q)-f(p)}{|pq|}> |\nabla_p f|-\eps.$$
Now choose $q_n\in A_n$ such that $q_n\to q$. 
If $|pq|$ is sufficiently small and $n$ is
sufficiently large,  the $\lambda$-concavity of $f_n$ then implies that
$$\liminf_{n\to \infty}\frac{d_{p_n}f_n(v_n)}{|v_n|}\ge |\nabla_p f|-2\eps \ \ \text{for}\ \  v_n=
\log_{p_n}(q_n)\in T_{p_n}(A_n).$$
Therefore,
$$\liminf_{n\to \infty} |\nabla_{p_n} f_n|\ge
|\nabla_p f|-2\eps \ \ \text{for any}\ \ \eps>0 ,$$
i.e., 
$$\liminf_{n\to \infty} |\nabla_{p_n} f_n|\ge
|\nabla_p f|. $$

\end{proof} 

\begin{lem}\label{lem:concave}Let $f$ be a $\lambda$-concave function,  $\lambda\ge 0$ and $\alpha(t)$ be an $f$-gradient curve, and let $\bar\alpha(s)$ be its reparameterization by arclength. Then $f\circ\bar\alpha$ is  $\lambda$-concave.
\end{lem} 

\begin{proof}

$$(f\circ\bar\alpha)^+(s_0)=|\nabla_{\bar\alpha(s_0)}f|
\ge
 \frac{d_{\bar\alpha(s_0)}f(\log_{\bar\alpha(s_0)}(\bar\alpha(s_1))}
 { |\bar\alpha(s_1)\,\bar\alpha(s_0)|}
 \ge$$
$$
\ge\frac{f(\bar\alpha(s_1))-f(\bar\alpha(s_0))-\lambda |\bar\alpha(s_1)\,\bar\alpha(s_0)|^2/2 }{|\bar\alpha(s_1)\,\bar\alpha(s_0)|}
\ge 
\frac{f(\bar\alpha(s_1))-f(\bar\alpha(s_0))}{ s_1-s_0}-\lambda|\bar\alpha(s_1)\,\bar\alpha(s_0)|/2.$$
Since $\frac{|\bar\alpha(s_1)\,\bar\alpha(s_0)|}{(s_1-s_0)}\to 1$
 as $s_1\to s_0+$, it follows that $f\circ\bar\alpha$ is  $\lambda$-concave.

\end{proof}  





\subsection{ Gradient push}
Let $f$ be a $\lambda$-concave function on an Alexandrov space $A$. 
Consider the map $\Phi_f^T:A\to A$ defined as follows:
$\Phi_f^T(x)=\alpha_x(T)$, where $\alpha_x:[0,\infty)\to A$ is the $f$-gradient curve with $\alpha_x(0)=x$.
The map $\Phi_f^T$ is called 
{\it $f$-gradient push at time $T$}. From Proposition~\ref{prop:gradlip}  it is clear that  $\Phi_f^T$ is an $\exp(\lambda T)$-Lipschitz map. Next we want to prove that this map behaves nicely under Gromov-Hausdorff-convergence.

\begin{thm}\label{thm:grconv} 
Let $A_n \buildrel \text{GH} \over \rightarrow A$ 
be a sequence of Alexandrov spaces with curvature $\ge k$ 
which converges to an Alexandrov space $A$.

Let $f_n\to f$, where $f_n:A_n\to \RR$ is a sequence of $\lambda$-concave functions and $f:A\to \RR$.

Then  $\Phi_{f_n}^T\to\Phi_f^T$.

\end{thm}

Theorem~\ref{thm:grconv}  immediately follows  from the following Lemma:

\begin{lem}Let $A_n \buildrel \text{GH} \over \rightarrow A$ 
be a sequence of Alexandrov spaces with curvature $\ge k$ 
which converges to an Alexandrov space $A$.

Let $f_n\to f$, where $f_n:A_n\to \RR$ is a sequence of $\lambda$-concave functions and $f:A\to \RR$.
 
Let $\alpha_n: [0,\infty) \to A_n$ be the sequence of 
$f_n$-gradient curves with $\alpha_n(0)=p_n$ and 
let $\alpha: [0,\infty) \to A$ be the $f$-gradient curve with $\alpha(0)=p$.

Then $\alpha_n\to\alpha$.
\end{lem}


\begin{proof}
We may assume without loss of generality  that $f$ has no critical points.
(Otherwise consider instead the  sequence $A'_n=A_n\times\RR$ with $f'_n(a\times x)=f_n(a)+x$.)

Let $\bar\alpha_n(s)$ denote the reparameterization of $\alpha_n(t)$ 
by arc length.
Since all $\bar\alpha_n$ are $1$-Lipschitz, 
 we can choose a converging subsequence from 
any subsequence of $\bar\alpha_n$. 
Let $\bar\beta:[0,\infty)\to A$ be its limit.

Clearly, $\bar\beta$ is also 1-Lipschitz and hence $|\bar\beta^+|\le 1$.
Therefore, by Lemma \ref{lem:gradcon},
$$\lim_{n\to\infty}f_n\circ\bar\alpha_n|_a^b=
\lim_{n\to\infty}\int_a^b|\nabla_{\bar\alpha_n(s)} f_n|ds\ge $$
$$\ge\int_a^b|\nabla_{\bar\beta(s)} f|\ge 
\int_a^b d_{\beta(t)} f(\beta^+(t))=f\circ\beta|_a^b$$

On the other hand, since $\bar\alpha_n\to\bar\beta$ and $f_n\to f$ we have
$f_n\circ\bar\alpha_n|_a^b \to f\circ\bar\beta|_a^b$. 
Therefore, in both of these inequalities in fact equality holds. 

Hence, $|\nabla_{\bar\beta(s)} f|= \lim_{n\to\infty} |\nabla_{\bar\alpha_n(s)} f_n|,\ \  
|\bar\beta^+(s)|= 1$
and the directions of $\bar\beta^+(s)$ and  $\nabla_{\bar\beta(s)} f$ 
coincide almost everywhere.
This implies that $\bar\beta(s)$ is a gradient curve reparameterized by 
arc length. 
In other words, if $\bar\alpha(s)$ denotes the
reparameterization of $\alpha(t)$ 
by arc length, 
then $\bar\beta(s)=\bar\alpha(s)$ for all $s$. 
It only remains to show that the original 
parameter $t_n(s)$ of $\alpha_n$ converges to the original 
parameter $t(s)$ of $\alpha$.

Notice that $|\nabla_{\bar\alpha_n(s)} f_n|dt_n=ds$ or 
$dt_n/ds=ds/d(f_n\circ\bar\alpha_n)$. 
Likewise, $dt/ds=ds/d(f\circ\bar\alpha)$. 
Then the convergence $t_n\to t$ follows from the $\lambda$-concavity of 
$f_n\circ\bar\alpha_n$ (see Lemma~\ref{lem:concave}) 
and the convergence $f_n\circ\bar\alpha_n\to f\circ\bar\alpha.$
\end{proof}

\begin{rmk}(A few words about the name ``gradient push'') 

At first sight, it might look strange why we call 
``gradient push'' something which is normally referred to 
as the gradient flow. 
The reason for this name stems 
from the relation of this notion to
moving furniture inside a room by pushing it around. 
If a piece of furniture is located 
in the middle of a room, 
one  can push it to any other place in the room.
But as soon as it is pushed 
to a wall one cannnot push it
back to the center; and once it is pushed into a corner one 
cannot push it anywhere. 
The same is true for the gradient push in an Alexandrov space, 
where the role of walls and corners is played by  extremal subsets.
\end{rmk}

\subsection{Gradient balls}

\ 

Let $A$ be an Alexandrov space and let $S\i A$ be a subset of $A$. A function $f:A\to \RR$  which can be represented as 
$$f=\sum_i \theta_i \frac{\dist^2_{a_i}}{2}\ \ \text{with }\ \  \theta_i\ge 0,\ \  \sum_i \theta_i=1\ \ \text{and}\ \ a_i\in S$$
will be called {\sl cocos-function with respect to $S$}
(where ``cocos'' stands for 
{\bf co}nvex {\bf co}mbination of {\bf s}quares of distance functions). 
A broken gradient curve for a collection of such functions
will be called cocos-curve with respect to $S$.
 
For $p\in A$ and $T,r\in \RR_+$, 
let us define  ``the gradient ball with center $p$ 
and radius $T$ with respect to $B_r(p)$'', $\beta^r_T(p)$, 
as the set of all end points of cocos-curves with respect to $B_r(p)$
that start at $p$ 
with total time $\le T$.

\begin{lem}\label{lem:gradb}\ 

\begin{enumerate}[(I)]
\item There exists $T=T(m)\in \RR_+$ 
such that for any $m$-dimensional Alexandrov space $A$ with curvature $\ge -1$ and any $q\in A$ there is a point $p\in A$ 
such that
\begin{enumerate}[(i)]
\item $|pq|\le 1$, and
\item $B_1(p)\i \beta^{1}_T(p)$.
\end{enumerate}
\item There exists $T'=T'(m)\in \RR_+$
such that the following holds.
Let $A$ be an Alexandrov space
which is a quotient $A=\tilde A/\Gamma$
of an  $m$-dimensional Alexandrov space $\tilde A$ with curvature $\ge -1$ 
by a discrete action of a group of isometries $\Gamma$.
%which acts properly discontinuously 
Let $q\in A$ and $p=p(q)\in A$
be as in part I above.

Then
for any lift $\tilde p \in \tilde A$ of $p$ one has that
$B_1(\tilde p)\i \beta1_{T'}(\tilde p)$.
\end{enumerate}
\end{lem}

\begin{proof}
The proof is similar to the construction of a strained point in an  Alexandrov space (see~\cite{BGP}). 
 
Set $\delta=10^{-m}$.
Take $a_1=q$ and take $b_1$ to be a farthest point from $a_1$ in the closed ball $\bar B_1(a_1)$.
Take $a_2$ to be a midpoint of $a_1b_1$ and let  $b_2$ be 
a farthest point from $a_2$ 
such that $|a_1b_2|=|a_1a_2|$ and $|a_2b_2|\le\delta|a_1b_1|$, etc.
On the $k$-th step we have to take $a_{k}$ to be a midpoint of
$a_{k-1}b_{k-1}$
and $b_{k}$  
to be a farthest point from $a_{k}$
such that $|a_ib_{k}|=|a_ia_{k}|$ for all $i< k$  and 
$|a_{k}b_{k}|\le\delta |a_{k-1}b_{k-1}|$.

After $m$ steps, take $p$ to be a midpoint of $a_mb_m$.
We only have to check that we can find $T=T(m)$
such that $\beta1_T(p)\supset B_1(p)$.

Let $t_i$ be the minimal time such that $B_{|a_ib_i|/\delta^m}(p)\i \beta1_{t_i}(p)$. 
Then one can take $T=t_1$.
Therefore it is enough to give estimates for $t_m$ and $t_{k-1}/t_k$ only in terms of $\delta$ and $m$.
Looking at the ends of broken gradient curves 
starting at $p$ 
for the functions $\dist^2_{p}/2$, $\dist^2_{a_i}/2$ 
and $ \dist^2_{b_i}/2$, we easily see that $t_n\le 1/\delta^m$.
Now, looking 
at the ends of broken gradient curves 
starting at $ B_{|a_{k-1}b_{k-1}|/\delta^m}(p)$ for 
the functions $\dist^2_{p}/2$, $\dist^2_{a_i}/2$ 
and $ \dist^2_{b_i}/2$, 
we have that $t_{k-1}/t_{k}\le 1/\delta^m$.
Therefore $t_1\le 1/\delta^{m2}=10^{-m3}$. 
This finishes the proof of part (I).

For part (II), 
notice that 
\begin{enumerate}[$*$]
\item for any $r,t>0$ we have
$\beta^r_t(p)\i B_{re^t}(p);$
\item if $\beta^r_t(p)\supset B_{\rho}(p)$, then
$\beta^r_{t+\tau}(p)\supset B_{\rho e^\tau}(p);$
\item if $\rho=|px|$ and $x\in \beta^{r+\rho}_t(p)$, 
then $\beta^{r}_\tau(x)\i \beta^{r+\rho}_{t+\tau}(p)$.
\end{enumerate}
Take $\eps=e^{-T}/4$ and apply part (I) of the lemma to $\frac1\eps\tilde A$
to find a point $p'\in \tilde A$ such that $|\tilde pp'|\le \eps$ 
and $B_{\eps}(p')\i \beta^{\eps}_{T}(p')\i \tilde A$. 
Then for some deck transformation $\gamma$ 
we have $\gamma p'\in \beta^\eps_T(p)\i B_{\eps e^T}(p)$. 
Therefore it holds that $\gamma p'\in B_{1/2}(\tilde p)$. 
Hence, taking 
$$T'=2T+1/\eps=2T+4e^{T},$$
we obtain
$$\beta1_{T'}(p)\supset \beta^\eps_{T+1/\eps}(\gamma p')
\supset B_1(\tilde p).$$
\end{proof}


\section{Nilpotency of almost nonnegatively curved manifolds}\label{sec:nilp}


In this section we prove Theorem~\ref{intro:main1}.

\subsection{Short basis.} We will use the following construction due to Gromov.

Given an Alexandrov space $A$ with a marked point $p\in A$,  and  a group $\Gamma$ acting discretely on $A$ one can define a short basis of the action of $\Gamma$ at $p$ as follows:

For $\gamma\in \Gamma$ define
$|\gamma|= d(p,\gamma(p))$.
Choose $\gamma_1\in \Gamma$ 
with  the minimal norm in $\Gamma$.
Next choose $\gamma_2$ to have 
minimal norm in $\Gamma\backslash <\gamma_1>$. 
On the $n$-th step choose 
$\gamma_n$ to have minimal norm in 
$\Gamma\backslash <\gamma_1,\gamma_2,...,\gamma_{n-1}>$.
The sequence $\{\gamma_1,\gamma_2,...\}$ 
is called a {\it short basis} of $\Gamma$ at $p$. 
In general, the number of elements of a short basis can be finite or infinite.
In the special case of the action of the fundamental group $\pi_1(A,p)$  on the universal cover of $A$
 one speaks of the short basis of  $\pi_1(A,p)$.
 
It is easy to see that for a short basis 
$\{\gamma_1,\gamma_2,...\}$ of the fundamental group of an Alexandrov space $A$
the following is true:
\begin{enumerate}
\item If $A$ has diameter $d$ then $|\gamma_i|\le 2d$.
\item If $A$ is compact then $\{\gamma_i\}$ is finite.
\item For any $i>j$ we have $|\gamma_i|\le |\gamma_j^{-1}\gamma_i|$.
\end{enumerate}
The third property implies that if 
$\tilde p \in \tilde A$ 
is in the preimage of $p$ in the universal cover $\tilde A$ of $A$
and $\tilde p_i=\gamma_i(\tilde p)$, then 
$$|\tilde p_i\tilde p_j|\ge 
\max\{|\tilde p\tilde p_i|,|\tilde p\tilde p_j|\}.$$

As was observed by Gromov, if $A$ is an Alexandrov space with curvature $\ge \kappa$ and diameter $\le d$, the last inequality implies that $\angle \tilde p_i \tilde p \tilde p_j>\delta=\delta(\kappa,d)>0$. 
This yields an upper bound on the number of elements
of a short basis in terms of $\kappa,d$ and the dimension of $A$.

\subsection{} In the proof, we will use the following simple observation. Let $G$ be a group acting on an abelian group $V$ via a representation
$\rho\co G\to \Aut(V)$.
It is obvious from the definition that the action  of $G$ on  $V$ is nilpotent iff the actions of $G$ on $Tor(V)$ and $V/Tor(V)$ are nilpotent.
It easily follows from Engel's theorem that if $V$ 
is finitely generated 
and if for any $g\in G$  all eigenvalues of the induced action of $g$ on $V\otimes \RR$ are equal to 1, then $G$ contains
 a finite index subgroup whose action on $V$ is nilpotent. 


Let $M$ be an almost nonnegatively curved manifold. 
Let us denote by $M_n=(M,g_n)$, $n\in \NN$,
a sequence of Riemannian metrics on $M$ such that $K_{M_n}\ge -1/n$ 
and diam$(M_n)\le 1/n$. 
Let us denote by $\tilde M$ the universal covering of $M$, and by  
$\tilde M_n\to M_n$ 
the universal Riemannian covering of $M_n$ 
(i.e., $\tilde M_n$ is 
$\tilde M$ equipped with the pullback of the Riemannian metric $g_n$).


\begin{klem}\label{l:key} Given $\eps>0$ and $r_2>r_1>0$,
let
$\tilde M_n\supset B_{r_2}(p_n)\supset B_{r_1}(p_n)$.
Then, for $n$ sufficiently large, 
there is a  $(1+\eps)$-Lipschitz map 
$\Phi_n:B_{r_2}(p_n)\to B_{r_1}(p_n)$ 
which is homotopic to the identity on $B_{r_2}(p_n)$.
\end{klem}

\begin{proof} Fix $R>\!\!>r_2$ 
(here $R>1000(1+1/\eps)r_2$ will suffice). 
Notice that as $n\to\infty$, we have that $B_R(p_n)\to B_R\i \RR^q$. 
Choose a finite $R/1000$-net $\{a_i\}$ of $\partial B_R\i \RR^q$.
Let $a_{i,n}\in M_n$ be sequences such that $a_{i,n} \to a_n$. 
Consider the sequence of functions 
$f_n:M_n\to \RR$ with $f_n=\min_i\text{dist}^2_{a_{i,n}}$.

For large $n$, the functions $f_n$ are $2$-concave in $B_R(p_n)$, 
so that, in particular, the gradient pushs
$\Phi_{f_n}^T|_{B_{r_2}(p_n)}$ are $e^{2T}$-Lipschitz.
Moreover, if $\xi_x$ denotes 
the starting vector of a unit speed shortest geodesic from $x$ to $p_n$, 
then  for any $x\in B_{r_2}(p_n) \backslash B_{r_1}(p_n)$ 
we have $\langle\xi_x,\nabla f \rangle\ge R/2$. 
Therefore, if $T=2r_2/R$, then $\Phi_{f_n}^T(B_{r_2}(p_n) )\i B_{r_1}(p_n)$.
Thus $\Phi_n=\Phi_{f_n}^{2r_2/R}$ provides a 
$4r_2/R$-Lipschitz map $B_{r_2}(p_n)\to B_{r_1}(p_n)$, 
and it is $(1+\eps)$-Lipschitz if one chooses $R$ sufficiently large. 
\end{proof}

\begin{cor}\label{cor:norm} Let $M$ be almost nonnegatively curved manifold.
Let $h:\pi_1(M)\to Aut(H^*(\tilde M,Z)/tor)$ 
be the natural  action of $\pi_1(M)$ on $(H^*(\tilde M)$.
Then there is a sequence of norms $|\!|*|\!|_n$ on 
$H^*(\tilde M,\ZZ)/tor$  
such that the following holds.
Given any $\eps>0$, there is $n\in \ZZ_+$ 
such that for any  $\gamma\in \pi_1(M)$ 
with $|\gamma|_n\le 2diam(M_n)$ we have  $|\!|h(\gamma)|\!|_n\le 1+\eps$.

\end{cor}

\begin{proof} \cite[theorem 0.1]{FY} and 
Yamaguchi's fibration theorem ~\cite{Yam} imply 
that if $n$ is sufficiently large, 
for any fixed $r\in \RR_+$
we have that for any $p_n\in \tilde M_n$ 
the inclusion map $B_r(p_n)\to \tilde M_n$ is a homotopy equivalence.

Let $|\!|*|\!|_{n,r}$ denote the $L_\infty$-norm 
on differential forms on $B_{r}(p_n)\i \tilde M_n$. 


Fix $r_2>r_1>0$. 
If $\omega$ is a differential form on  $B_{r_1}(p_n)\i M_n$ 
and $n$ is sufficiently large,  Lemma~\ref{l:key} implies that 
$$|\!|\Phi^*_n(\omega)|\!|_{n,r_2}\le(1+\eps)|\!|\omega |\!|_{n,r_1} 
\ \ \text{and}\ \ 2\text{diam}(M_n)\le r_2-r_1.$$

If now $\omega$ is a form on 
$B_{r_2}(p_n)\in \tilde M_n$ and 
$\gamma\in \pi_1(M)$ 
such that 
$$|\gamma|_n=|p_n\,\gamma(p_n)|\le 2\text{diam}(M_n)\le  r_2-r_1,$$ 
then $B_{r_1}(p_n)\i B_{r_2}(\gamma(p_n))\i \tilde M_n$, whence 
$$|\!|\Phi^*_n(\gamma^*(\omega))|\!|_{n,r_2}\le 
(1+\eps)|\!|\gamma^*(\omega)|\!|_{n,r_1}\le
(1+\eps)|\!|\omega|\!|_{n,r_2}.$$

Thus, for the induced norms on the de Rham cohomology 
of $\tilde M$ (and on its integral subspace $H^*(\tilde M,\ZZ)/tor)$)  we have
$$|\!|[\gamma^*(\omega)]|\!|_{n,r_2}\le 
(1+\eps)|\!|[\omega]|\!|_{n,r_2}.$$

Therefore the sequence of norms $|\!|*|\!|_n=|\!|*|\!|_{n,r_2}$ satisfies the conditions of the Corollary.
\end{proof}


\begin{lem}\label{lem:eigen} 
There exists a constant $N=N(n,k)\in \ZZ_+$ such that the following holds.
If $G$ is a subgroup of $GL(n,\ZZ)$ and $S$ is a set of generators of $G$
with $\#(S)\le k$ such that the eigenvalues of each element of
$S^N$ are all equal to $1$ in absolute value, 
then the same is true for the eigenvalues of all
elements of $G$.
\end{lem}






\begin{proof}

Let $B$ be the set of all matrices in $GL(n,\Z)$
for which all of their eigenvalues are equal to 1 in absolute value.
Since the characteristic polynomials of such matrices 
are uniformly bounded and have integer coefficients, 
there are only finitely many of them. 
Let $\bar{B}$ be the Zariski closure of $B$ in the set of all real 
$n\times n$ matrices. By the above, 
all elements of $\bar{B}$ satisfy that the absolute values of all 
of their eigenvalues are equal to 1.

Consider now the space $V=\RR^{kn2}$ of 
$k$-tuples of real $n\times n$ matrices.

Consider a  collection of matrices $(M_1,M_2,...,M_k)\in V$,
where $M_i\in GL(n,\RR)$.
Let $F_k$ be a free group on $k$ generators,
generated by $S=\{\gamma_1,\gamma_2,...,\gamma_k\}$,
and let $h:F_k\to GL(n,\RR)$ be the 
homomorphism defined by $h(\gamma_i)=M_i$.
The property that for any $\gamma\in F_k$ $h(\gamma)$
be an element of $\bar{B} $ 
then describes an algebraic subset $A_\gamma\i V$.

The  intersection $A=\cap_{\gamma\in F_k}A_\gamma$ 
is also  algebraic, and therefore there is a finite number $N=N(n,k)$ 
such that for $S^N\i  F_k$,
$A=\cap_{\gamma\in S^N}A_\gamma$.
\end{proof} 

\begin{lem}\label{lem:eigen1} 
Let $\Gamma$  be a subgroup of $GL(n,\ZZ)$ such that 
the eigenvalues of each element of $G$
are equal to $1$ in  absolute value.
% of all eigenvalues of $\gamma$ are equal $1$. 
Then $\Gamma$ contains a   subgroup  $\Gamma'$ of finite index
%such that for  any $a\in \Gamma'$  
all of whose  elements have eigenvalues equal to $1$.
\end{lem}

\begin{proof}
Let $G$ denote the Zariski closure of $\Gamma$ in  $GL(n,\RR)$ . 
Then $G$, being an algebraic group,
is a Lie group with finitely many components. 
Let $G_\circ$ be the identity  component of $G$. 
By the same argument as in the proof of the previous lemma,
the set of all characteristic polynomials 
of the elements of $G$ is finite.
Therefore the characteristic polynomial
of any element of $G_\circ$ 
is identically equal to $(x-1)^n$.

Therefore, the subgroup $\Gamma'=\Gamma\cap G_\circ$ 
satisfies all conditions of the Lemma. 
\end{proof}
\begin{comment}
\begin{rmk}
Alternatively, one can take $\Gamma'$ to be the kernel of composition of homomorphisms 
$\Gamma\to GL(n,\ZZ)\to GL(n,\ZZ/3\ZZ)$, 
this way we get an effective bound  $$[\Gamma:\Gamma']\le 3^{n2}.$$
REFERENCE?
\end{rmk}
\end{comment}

\begin{proof}[Proof of Theorem~\ref{intro:main1}]
Let $M$ be an almost nonnegatively curved manifold. 
Denote, as usual, by $M_n=(M,g_n)$, $n\in \NN$,
a sequence of Riemannian metrics on $M$ such that $K_{M_n}\ge -1/n$ 
and diam$(M_n)\le 1/n$, 
by $\tilde M$ the universal covering of $M$, and by  
$\tilde M_n\to M_n$ 
the universal Riemannian covering of $M_n$.
 
After passing to a finite cover of $M$, by~\cite{FY}
we may assume that
 $\pi_1( M)$ is nilpotent. 

Fix $p\in M$ and 
let $\{\gamma_{i,n}\}$ be a short basis of $\pi_1(M_n,p)$.
Then, if $n$ is sufficiently large, 
the short basis $\{\gamma_{i,n}\}$ 
has at most $k=k(\dim M)$ elements and 
its elements satisfy  
$|\gamma_{i,n}|_n\le 2/n$ for every $i$.
Moreover, 
Corollary~\ref{cor:norm}  implies 
that given $\eps>0$, for all large $n$ and every $i$ 
we have $|\!|h(\gamma_{i,n})|\!|_n <
1+\eps$ and $|\!|h(\gamma^{-1}_{i,n})|\!|_n<1+\eps$.

Take $N=N(k,m)$ as in Lemma~\ref{lem:eigen}. 
One can choose $\eps>0$ 
so small that 
if $p$ is a polynomial with integer coefficients 
for which all of its roots have  absolute 
values lying between  $1/(1+\eps)^N$ and $(1+\eps)^N$,
then  all roots of $p$ have absolute values equal to $1$. 
This follows from the fact 
that the total number of integer polynomials 
all of whose roots are contained in a fixed bounded region is finite.


Set $S_n:=\{\gamma_{i,n}\}$. 
Then for any $\gamma\in S_n^N$ we have 
$|\!|h(\gamma)|\!|_n<(1+\eps)^N$ and 
$|\!|h(\gamma^{-1})|\!|_n<(1+\eps)^N$.
Therefore the absolute values of all eigenvalues 
lie between $1/(1+\eps)^N$ and $(1+\eps)^N$.  
Since the characteristic polynomial of $h(\gamma)$ 
has integer coefficients, 
the absolute values of all the eigenvalues of $h(\gamma)$ are 
in fact equal to $1$.

Apply now Lemma~\ref{lem:eigen}. It follows
that for any $\gamma\in \pi_1(M)$ 
the absolute values of all  eigenvalues of $h(\gamma)$ are equal to $1$.

Then Lemma~\ref{lem:eigen1}  implies that after passing to a finite 
cover $M'$ of $M$, 
for any $\gamma\in \pi_1(M')$ 
all eigenvalues of $h(\gamma)$ are equal to $1$. 
By Engel's theorem, one can choose an integral basis 
of $H^*(\tilde M,\RR )$ 
such that the action of $\pi_1(M)$ 
on $H^*(\tilde M, \ZZ)/tor$ is given by upper triangular matrices.

Therefore, by passing to a finite cover $M''$ of $M'$, 
we can assume that the action of $\pi_1(M'')$ 
on $H^*(\tilde M,\ZZ )$  (and  on $H_*(\tilde M,\ZZ )$)  is nilpotent.

Recall (see e.g. ~\cite[2.19]{HMR}) 
that a connected CW complex with nilpotent fundamental group 
is nilpotent if and only if the action of
its fundamental group on the homology of its universal cover is nilpotent. 
Thus $M''$ is a nilpotent space, whence the proof of Theorem~$A$
is complete.
\end{proof}








\section{C-nilpotency of the fundamental group}\label{sec:c-nilp}

In this section we will prove Theorem~\ref{intro:main2}. It will follow
from the following somewhat stronger result.




\begin{thm}\label{thm:cnilp}
For any integer $m$ there exist constants $\e(m)>0$  and $C(m)>0$ such that 
the following holds. 
If
$M^m$ is a closed smooth $m$-manifold which admits 
a Riemannian metric with 
$\sec(M^m)> -\e(m)$ and $\diam(M^m)< 1$,
then the fundamental group of $M^m$ is $C(m)$-nilpotent, i.e.,
$\pi_1(M^m)$ contains a nilpotent subgroup of index $\le C(m)$.
\end{thm}
\begin{rmk} %As a matter of fact,
The proofs of Theorems~\ref{intro:main1} and ~\ref{intro:main3} show
that
corresponding versions of those results
also do hold when these theorems are reformulated 
in a fashion similar to Theorem~\ref{thm:cnilp}.
\end{rmk}
By an argument by contradiction, 
Theorem~\ref{thm:cnilp} follows
from the following  statement:

Given a sequence of Riemannian $m$-manifolds $(M_n,g_n)$ 
with diameters $\text{diam}(M_n,g_n)\le 1/n$ 
and sectional curvatures $K_{g_n}\ge -1/n$, 
 one can find $C\in \RR$  such that 
$\pi_1(M_{n})$ is $C$-nilpotent for all sufficiently large $n$.

To prove this statement, 
we will  make use of the following two  algebraic lemmas. 

Recall that 
the group of outer automorphisms $\Out(G)$ of a group $G$ 
is defined as the quotient of its automorphism group 
$\Aut(G)$ by the subgroup of inner automorphisms $\Inn(G)$.

 
\begin{lem}[\it A characterization of $C$-nilpotent groups]\label{lem:cnilp} 
Let 
$$\{1\}=G_l\subseteq \ldots\subseteq G_1\subseteq G_0=G$$
  be a sequence of groups satisfying the following properties:
  
  For any $i$

(i) $G_i \unlhd G$ is normal in $G$;
  
(ii) the image of the conjugation homomorphism 
$h_i: G\to \Out(G_i/G_{i+1})$ is finite of order at most $C_i$;
  
(iii) $G_i/G_{i+1}$ contains an abelian subgroup $A_i$ of index  $\le c_i$.

\noi Then $G$ contains a nilpotent subgroup $N$ of index at most 
$C=C(c_1,\ldots, c_l, C_1,\ldots, C_l)$, where $N$ is of 
nilpotency class $\le l$.
\end{lem}
\begin{proof}
First of all, notice that 
property (i) assures that the 
objects described in parts (ii) and (iii) of the lemma are well-defined.

Denote $\G_i=G_i/G_{i+1}$.

Let $H=\cap\ker{h_i}$. Notice that $[G:H]\le \prod_iC_i$ and the image of $H$ under

 the conjugation homomorphism $f_i: G\to \Aut(\G_i)$  lies in $\Inn(\G_i)$, i.e $f_i|_H\co H\to  \Inn(\G_i)$.
 
 By passing to a subgroup we can assume that $A_i\unlhd \G_i$ is normal of index $\le C(c_i)$ (we can take $C(c_i)=c_i!$).  By increasing $A_i$ if necessary we can assume that $A_i$ contains the center of $\G_i$.
 
 Let $Z_i$ be the image of $A_i$ under the projection map $\pi\co \G_i\to \Inn(\G_i)$. Clearly $[ \Inn(\G_i): Z_i]\le c_i$ and 
 $Z_i\unlhd  \Inn(\G_i)$ is normal.
 

 Let $N=H\cap\cap_if_i^{-1}(Z_i), N_i=N\cap G_i$. Then $[G:N]\le C=C(c_1,\ldots, c_l, C_1,\ldots, C_l)$ and $N$ satisfies:
 
 
  For any $i$
  
  \begin{enumerate}[(i')]
\item
 $N_i \unlhd N$ is normal in $N$;

\item
$N_i/N_{i+1}$ is in the center of $N/N_{i+1}$
\end{enumerate}
 i.e $N$ is nilpotent of nilpotency length $\le l$.
 
 Condition (i') is obvious so we only need to check (ii').
 
 To see (ii') observe that by construction the image of the conjugation action $N\to \Aut(\G_i)$ lies in $\Inn(\G_i)$. Moreover, in fact it lies in $\pi(A_i)$ and as such it acts trivially on $A_i$. Lastly observe that  $N_i/N_{i+1}\subset A_i$.
 
 Indeed, by construction, for any $g\in N_i/N_{i+1}\subset \G_i$ there is $a\in A_i$ such that $\pi(g)=\pi(a)$. Therefore, $g=az$ for some $z$ in the center of $\G_i$. By our assumption on $A_i$ this means that $g\in A_i$.

 Thus $N$ acts trivially on $N_i/N_{i+1}$ which means that $N$ is nilpotent and $G$ is $C$-nilpotent.
 \end{proof}

\

\begin{tlem}[\it A characterization of finite actions]\label{lem:triv} 
If $S$ is a finite set of generators of a group $G$ with $S^{-1}=S$, 
and $h:G\to H$ is a homomorphism with $|h(S^n)|< n$ for some $n>0$,
then $h(S^n)=h(G)$ and, in particular, $|h(G)|< n$.
\end{tlem}







Let now $\Gamma$ be a group which acts discretely 
by isometries on an Alexandrov space $A$ with curvature $\ge -1$. 
Choose a marked point $p\in A$.
Assume that $\{\gamma_1,\gamma_2,\ldots,\gamma_n\}$ 
is a finite short basis of $\Gamma$ at $p$, 
and that $\theta\le|\gamma_i|\le 1$, 
where $|\gamma|\overset{\rm def}{=}|p\gamma(p)|$.
Let $\#(R)$ 
denote the number of delements $\gamma\in\Gamma$ with $|\gamma|\le R$.
The  Bishop-Gromov inequality implies that 
$$\#(R)\le v^m_{-1}(R)/v^m_{-1}(\theta),$$
where $m=\dim A$ and $v^m_{-1}(r)$ 
is the volume of the ball of radius $r$ 
in the $m$-dimensional simply connected space form of curvature $-1$.
Therefore, 
if $\#\#(L)$ denotes the number of homomorphisms 
$h:\Gamma\to \Gamma$ with norm $\le L$ 
(i.e., the number of homomorphisms 
for which it holds that for any $\gamma\in\Gamma$ 
one has that $|h(\gamma)|\le L|\gamma|$), 
then
\begin{equation}\label{e:hombd} 
\#\#(L)\le \#(L)^n\le \bigl[\frac{v^m_{-1}(L)}{v^m_{-1}(\theta)}\bigr]^n.
\end{equation}



\subsection{The construction}\label{sec:constr}

\ 

For $n\to\infty$, the manifolds $M_n$ clearly converge to a point $=:A_0$. 

Set $M_{n,1}:=M_n$ and $\vartheta_{n,1}:=\diam M_{n,1}$.

Rescale now $M_{n,1}$ by $\frac{1}{\vartheta_{n,1}}$ 
so that $\text{diam}(M_{n,1}/\vartheta_{n,1})=1$. 
Passing to a subsequence if necessary, one has  
that the manifolds $\frac{1}{\vartheta_{n,1}}M_{n,1}$
converge to $A_1$,
where $A_1$ 
is a compact nonnegatively curved Alexandrov space with diameter $1$. 

Now choose a regular point $\bar p_1\in A_1$, 
and consider distance coordinates around $\bar p_1\in U_1\to \RR^{k_1}$, 
where $k_1$ is the dimension of $A_1$. 
The distance functions can be lifted 
to $U_{n,1}\i \frac{1}{\vartheta_{n,1}}M_{n,1}$.

Let  $M_{n,2}$  be the level set of $U_{n,1}\to \RR^{k_1}$ 
that corresponds to $\bar p_1$. 
Clearly, $M_{n,2}$ is a submanifold of codimension $k_1$.
Set $\vartheta_{n,2}:=\diam M_{n,2}$.
 
Passing again to a subsequence if necessary, one has
that the sequence $\frac{1}{\vartheta_{n,2}}M_{n,2}$ 
converges to some  Alexandrov space $A_2$.
As before, $A_2$ is a compact nonnegatively curved Alexandrov space 
with diameter 1. Set $k_2:=k_1+\dim A_2$. 
If one now chooses a marked point in $M_{n,2}$, 
then, as $n\to\infty$, 
$M_n/\vartheta_{n,2}$ converges to $A_2\times\RR^{k_1}$,
which is of some dimension $k_2>k_1$.

We repeat this procedure until, at some step, $k_l=m$.

\ 

As a result one obtains 
a sequence $\{A_i\}$ of compact nonnegatively curved Alexandrov spaces 
with diameter $1$ that satisfies
$$\text{dim} A_i=k_i-k_{i-1}, \hbox{\ \ so that 
\ \ }\sum_{i=1}^l\text{dim} A_i=m.$$
We also obtain 
a sequence of rescaling factors $\vartheta_{n,i}=\diam M_{n,i}$, 
and  a nested sequence of submanifolds 
$$\{p_n\}=M_{n,l}\i \cdots \i M_{n,2}\i M_{n,1}=M_n,$$
which in turn induces a sequence of homomorphisms
$$\{1\}=\pi_1(M_{n,l})\buildrel {\imath} 
\over \rightarrow \cdots\buildrel {\imath} 
\over \rightarrow \pi_1(M_{n,2})\buildrel {\imath} 
\over \rightarrow \pi_1(M_{n,1})=\pi_1(M_n).$$


Let $G_i:=G_i(n):=\imath^i\pi_1(M_{n,i})$.

For $n$ sufficiently large, 
the subgroups $G_i(n)$ are clearly those which are
generated by elements of  norm $\le 3\vartheta_{n,i}$. 
Equivalently, if one takes a short basis $\{\gamma_i\}$ of $G(n)$, 
then $G_i$ is the subgroup generated by all elements $\gamma_i$ 
such that $|\gamma_i|\le 3\vartheta_{n,i}$.

\ 



\subsection{ Limit fundamental groups of  Alexandrov spaces.}\label{fun-gr}

\ 

We will now define the ``limit'' or ``L-fundamental groups'' of 
the Alexandrov spaces $A_i$ constructed above.
This notion is similar to the notion of  the fundamental group of an orbifold.
However, we note in advance 
that the construction of the L-fundamental group does not only
depend on the spaces $A_i$, 
but also on the chosen rescaled subsequence of $M_n$. 
In fact, the following construction shows 
that the limit fundamental group of $A_i$, 
$\pi^L_1(A_i)$,
 is isomorphic to $\pi_1(M_{n,i},M_{n,i+1})$
for all sufficiently large $n$.
But, unlike $\pi_1(M_{n,i})$, the groups $\pi_1^L(A_i)$ will not depend on $n$.

{\it The limit fundamental groups of $A_i$.} Consider the converging sequence 
$(M_{n}/\vartheta_{n,i},p_{n})\to (A_i\times \RR^{k_{i-1}},\bar p_i\times 0)$
(here the interesting case is collapsing). 
Recall that 
$\bar p_i\in A_i$
is a regular point. 
Fix $\eps>0$ such that 
$\text{dist}_{\bar p_i}$ on $A_i$
does not have critical values in $(0,2\eps)$.
Take a sequence $R_n$ which converges very slowly to infinity 
(here 
we will need $R_n\vartheta_{n,i}/\vartheta_{n,i-1}\to 0$ and $R_n\to \infty$). 

Consider then 
a sequence of Riemannian coverings 
$\Pi\co (\tilde B_n,\tilde p_n)\to (B_{R_n}(p_n), p_n)$ of 
$B_{R_n}(p_n)\i {M}_n/\vartheta_{n,i}$ with 
$\pi_1(\tilde B_n,\tilde p_n)=\pi_1(B_\eps(p_{n}), p_n)$, 
where $B_\eps(p_{n})\i M_n/\vartheta_{n,i}$. 

After passing to a subsequence if necessary, 
the sequence $(\tilde B_n,\tilde p_n)$
converges to a nonnegatively curved Alexandrov space 
$\tilde A_i\times \RR^{k_{i-1}}$, 
where the space $\tilde A_i$ has the same dimension as $A_i$. 
Indeed, by construction it  contains an isometric copy of 
$B_\eps(p_{n,i})$, and therefore 
$$\dim \tilde A_i +k_{i-1}= 
\dim \lim_{i\to\infty} B_\eps(p_{n,i})=\dim A_i+k_{i-1}.$$
 
Let us show that for all sufficiently large $n$, 
$$\imath(\pi_1(M_{n,i+1}))\trianglelefteq\pi_1(M_{n,i}).$$
Assume that $\Pi(\tilde q_n)=\tilde p_n$ 
and that $\tilde q_n\to \bar q_n\in \tilde A_i$.
Connect $\bar p_n$ and $\bar q_n$ by a geodesic
which, by \cite{Ptr3}, only passes  through regular points. 
Note that in a small neighborhood  of this geodesic in $M_n$
 we have two copies of $M_{n,i+1}$, near $\tilde p_n$ and $\tilde q_n$.
Therefore, 
applying Yamaguchi's  Fibration Theorem 
in a small neighborhood  of this geodesic, 
we can construct a diffeomorphism from $M_{n,i+1}$ to itself. 
This  implies that for any loop $\gamma$ which after lifting 
connects $\tilde p_n \tilde q_n$, 
we have $\gamma^{-1}\imath\pi_1(M_{n,i+1})
\gamma\i \imath\pi_1(M_{n,i+1})$,  i.e., 
$\imath\pi_1(M_{n,i+1})\lhd\pi_1(M_{n,i})$ 
(for an alternative argumen see also ~\cite{FY}).


This easily yields that $A_i=\tilde A_i/\G_i$, 
where $\G_i$ is a group of isometries 
which acts %properly discontinuously 
discretely on $\tilde A_i$. 
The group $\G_i$ is denoted  by $\pi_1^L(A_i)$ 
(the {\it limit} or {\it L-fundamental group} of $A_i$).
This group  is clearly isomorphic to 
$$\pi_1(M_{n,i},M_{n,i+1})=\pi_1(M_{n,i})/\imath( \pi_1(M_{n,i+1}))$$ 
for all sufficiently large $n$, and
the space $\tilde A_i$ will be called the {\it universal covering} of $A_i$.

Since $\tilde A_i$ is nonnegatively curved  
and $A_i=\tilde A_i/\pi_1^L(A_i)$ is compact, 
by Toponogov's splitting theorem  
$\tilde A_i$ isometrically splits  
as $\tilde A_i=K_i\times \RR^{s_i}$, 
where $K_i$ is a compact Alexandrov space with curv$\ge 0$. 
Since $\pi_1^L(A_i)$ is a group of isometries 
that acts %properly discontinuously 
discretely on $\tilde A_i$, 
it follows that $\pi_1^L(A_i)$ is a virtually abelian group.

\ 

Consider now the corresponding series
$$\{1\}=G_l(n)\i\ldots\i G_1(n)\i G_0(n)=\pi_1(M_n).$$

The theorem then follows from the following 

\ 

\begin{lem}\label{l:claim} 
For all sufficiently large $n$, the series
$$\{1\}=G_l(n)\i\ldots\i G_1(n)\i G_0(n)$$
constructed above
satisfies the assumptions of Lemma~\ref{lem:cnilp} 
for numbers $C_i$ and $c_i$ which do not depend on $n$.
\end{lem}
 
We first prove the following 
\begin{slem}\label{lem:normal}
 Each subgroup $G_i(n)$ is normal in $G(n)$.
\end{slem}

\begin{proof}
We will show by reverse induction on $k$ 
that $G_i(n)\unlhd G_{k}(n)$ for any $k\le i$.
Let us assume that we already know that $G_i(n)\unlhd G_{k+1}(n)$.  
Since $$\imath\pi_1(M_{n,k+1})\unlhd\pi_1(M_{n,k}),$$ 
we know that $G_{k+1}(n)\unlhd G_{k}(n)$.
Consider the covering 
$\Pi_{k+1}\co (\tilde M_{n,k+1},\tilde p_{n,k+1})\to (M_n,p_n)$ 
with covering group $\G_{k+1}(n)$.



Clearly $(\tilde M_{n,k+1},\tilde p_{n,k+1})\to \RR^{s_i}$ 
for some integer $s_i$.
Applying Lemma~\ref{lem:gradb}, it follows 
that for any $a\in G$ with $|a|<1$ there is 
a cocos-curve $\gamma$ in $\tilde M_{n,k+1}$ 
with total time $T$ connecting 
$\tilde p_n$ and $a(\tilde p_n)$ in $\tilde M_{n,k+1}$. 
Then clearly $\gamma\sim ga$ for some $g\in G_{k+1}(n)$. 
Let us denote by $\Phi^T: \tilde M_{n,i}\to \tilde M_{n,i}$ the gradient push 
corresponding  to $\gamma$. 

Let $\gamma_j$ be a loop 
from the short basis of $G_i(n)$. 
As was mentioned in ~\ref{sec:constr}, 
if $n$ is large, then $\length\gamma_j\le 3\vartheta_{n,i}$.
Let us denote by  $\tilde\gamma_j$ a lift of $\gamma_j$ to $\tilde M_{n,i}$. 
Let $\tilde p_{n,j}\in \tilde M_{n,i}$ be its starting point. 
Since $[\gamma_j]\in G_i(n)$, we have that 
$\tilde\gamma_j$ is a loop in $\tilde M_{n,i}$. 
Consider then the loop
 $\gamma_j'=\Pi\circ\Phi^T\circ
\tilde\gamma_j$.  
Clearly, 
$$[\gamma_j]=a^{-1}g^{-1}[\gamma_j']ga , 
\ \ \text{or}\ \ [\gamma_j']=ga[\gamma_j]a^{-1}g^{-1}.$$


Now Proposition~\ref{prop:gradlip} implies that
$$\length(\gamma_j')\le \exp(2T)\length(\gamma_j).$$ 
Thus, for sufficiently large $n$,
$$ ga[\gamma_j]a^{-1}g^{-1} \in G_i(n),$$

and since $g\in G_i(n)\unlhd  G_{k+1}(n)$ it follows that
$$ a[\gamma_j]a^{-1} \in G_i(n),$$
i.e., $G_i(n)\unlhd  G_{k}(n)$.

\end{proof}



\begin{proof} [Proof of Lemma~\ref{l:claim}]

The group  
$$\pi_1^L(A_i)=\pi_1(M_{n,i},M_{n,i+1})=\pi_1(M_{n,i})/\imath(\pi_1(M_{n,i}))$$
is virtually abelian. 
Let $d_i$ be the minimal index of an abelian subgroup of $\pi_1^L(A_i)$. 
The epimorphism 
$\imath^i:\pi_1(M_{n,i})\to G_i$ 
induces an  epimorphism $\pi_1^L(A_i)\to G_i(n)/G_{i+1}(n)$.
Therefore, $G_i(n)/G_{i+1}(n)$ is $d_i$-abelian for all large $n$. 


Consider the covering 
$\Pi_i:\tilde M_{n,i}\to M_n$ with covering group $G_i(n)$, 
and let $\tilde p_{n,i}$ be a preimage of $p_n$. 
Clearly $(\tilde M_{n,i},\tilde p_{n,i})\to \RR^s{_i}$ 
for some integer $s_i$.
Applying Lemma~\ref{lem:gradb}, 
it follows  that for any $a\in G(n)$ with $|a|<1$ there is 
a cocos-curve $\gamma$ in $\tilde M_{n,i}$ which connects $p$ and $a(p)$. 
Then clearly $\gamma\sim ga$ for some $g\in G_i(n)$. 
Let us denote by $\Phi^T: \tilde M_{n,i}\to \tilde M_{n,i}$ 
the gradient push corresponding  to $\gamma$. 

Let $b\in G_i(n)$ and $\beta$ be a loop  representing $b$.
Let us denote by  $\tilde\beta$ a lift of $\beta$ to $\tilde M_{n,i}$. 
Let $\tilde p_{n,i}\in \tilde M_{n,i}$ be its starting point. 
Since $[\beta]\in G_i(n)$, 
we have that $\tilde\beta$ is a loop in $\tilde M_{n,i}$. 

Consider now the loop 
$\beta'=\Pi\circ\Phi^T\circ\tilde\beta$. Clearly, 
$$b=[\beta]=a^{-1}g^{-1}[\beta']ga , \ \ \text{or}\ \ [\beta']=gaba^{-1}g^{-1}.$$
Proposition~\ref{prop:gradlip}  then implies that
\begin{equation*}\label{e:1}\length(\beta')\le \exp(2T)\length(\beta)
\end{equation*}
Therefore, if $h_a:G_i(n)/G_{i+1}(n)\to G_i(n)/G_{i+1}(n)$ 
is induced by the conjugation $b\to aba ^{-1}$, 
then for any $a\in G(n)$ there is $g\in G_i(n)$ such that $|h_{ga}|\le \exp(2T)$. 

Let now $\delta_i$ be the minimal norm of the elements of $\pi_1^L(A_i)$,
where $\pi_1^L(A_i)$ acts on $\tilde A_i$. 
Then (\ref{e:hombd})  implies that the image of the 
action of $G(n)$ by conjugation in $Out(G_i(n)/G_{i+1}(n))$ is $C_i$-finite, 
where  $C_i$ depends only on $c_i$, $T$,
and $\delta_i$.
\end{proof}


\begin{proof} [End of the proof of Theorem~\ref{thm:cnilp}]
%This makes it possible to 
Apply now Lemma~\ref{lem:cnilp}
to obtain that $G(n)$ is $C$-nilpotent for $C= C(c_1,\ldots, c_l, C_1,\ldots, C_l)$, 
whence the proof of Theorem~\ref{thm:cnilp} is complete. %\hfill $\qed$
\end{proof}

\begin{rmk}
Theorem~\ref{thm:cnilp} can be reformulated as follows:
There exists a constant $\e(m)>0$ such that 
if $N^m$  is a Riemannian manifold 
which admits a discrete free  isometric action by a group $\Gamma$ 
such that $\sec(N)> -\e(m)$ and $\diam(N/\Gamma)<1$, 
then $\Gamma$ is $C(m)$-nilpotent.
\end{rmk}

As was pointed out to us by B.~ Wilking, 
in the above reformulation of Theorem~\ref{thm:cnilp} 
one can actually easily remove the assumption
that the $\Gamma$ action be free.

\begin{cor}\label{cor:nonfree}
There exists a constant $\e(m)>0$ such that
if $N^m$  is a Riemannian manifold 
which admits a discrete isometric action by a group $\Gamma$ 
such that $\sec(N)> -\e(m)$ and $\diam(N/\Gamma)<1$, 
then $\Gamma$ is $C(m)$-nilpotent.
\end{cor}
\begin{proof}
Let $\e=\e(m)$ be as provided by Theorem~\ref{thm:cnilp} 
and suppose $N$ satisfies the assumptions of the corollary for this $\e$.
Let $F$ be the frame bundle of $N$.
Then the action of $\G$ on $N$ lifts to a free isometric action on $F$.
As was observed in~\cite{FY}, 
using Cheeger's rescaling trick 
$F$ can be equipped with a $\G$ invariant metric 
satisfying $\sec(F)>-\e(m)$ and $\diam(F/\Gamma)<1$.   
Since the induced action of $\Gamma$ on $F$ is free, 
the claim of the corollary now follows from Theorem~\ref{thm:cnilp}.
\end{proof}




\section{Proof of the Fibration Theorem}\label{sec:fib}
Let $M$ be an almost nonnegatively curved manifold. 
Let us denote by $M_n=(M,g_n)$
a sequence of Riemannian metrics on $M$ such that $K_{M_n}\ge -1/n$ 
and diam$(M_n)\le 1/n$. 

Let us denote by $\tilde M$ the universal cover of $M$ and by  
$\tilde M_n\to M_n$ 
the universal Riemannian covering of $M_n$ 
(i.e., $\tilde M$ equipped with the pull back of the metric $g_n$ on $M$).

By \cite{FY}, passing to a finite cover 
%(applying Fukaya-Yamaguchi theorem~), 
we may assume that $\G=\pi_1(M)$ is a nilpotent group without torsion. 
Hence, to prove the topological part of Theorem $C$,
it is enough to show the following:

\begin{thm}\label{thm:fib}

Let $M$ be a closed almost nonnegatively curved $m$-manifold 
such that $\G=\pi_1(M)$ is a nilpotent group without torsion.
Then  $M$ 
is the total space of a fiber bundle
 \[
 F\to M\to N
 \]
where the base $N$ is a nilmanifold and the fiber $F$ is simply connected.
\end{thm}

The assumption on $\G$ implies that
we can fix a series 
$\G=\G_0\rhd\G_1\rhd\G_2\rhd\ldots\rhd\G_l=\{1\}$ 
such that $\G_i$ is normal in $\G$ and $\G_i/\G_{i+1}\cong \Z$.

Let us first us give an informal proof. 

\subsection{An informal proof of Theorem~\ref{thm:fib}} 
We use induction to construct the bundles  
$F_i\to M\overset{f_i}{\to} N_i$, where each $N_i$ is a 
nilmanifold with $\pi_1(N_i)=\G/\G_i$ 
and $\pi_1(F_i)\cong \G_i$. 
Since the base of induction is trivial, 
we are only interested in the induction step.

Fix $p\in N_i$,  
and let $F_i(p)$ be the fiber over $p$.
For any sufficiently large $n$ 
choose a subgroup $G_i=G_i(n)$ 
such that $\G_{i}\lhd G_i\lhd \G_{i+1}$ and 
$[\G_i: G_i]$ is finite, 
but large enough so that the cover  
$\bar{F}_i(p)$ of $F_i(p)$ 
corresponding to $G_i$ is Hausdorff close to a unit circle $S^1$.
 

Construct now a bundle map $\phi_p\co \bar F_i(p)\to S^1$
by lifting distance functions from $S^1$
%using 1-strainers coming from $S^1$ 
(This can be done by a slight generalization 
of a construction in ~\cite{FY}  and~\cite{BGP}). 
Let $\omega_p=d\phi_p$.

Then $\omega_p$
is a closed integral non-degenerate one-form on $F_i(p)$. 
Since deck transformations are isometries, 
after  averaging by $\Z_a$, where $a= [\G_i: G_i]$, 
we can assume that
$\omega_p$ is $\Z_a$-invariant.
Thus $\omega_p$ descends to a form on $F_i(p)$ 
which when integrated gives a bundle map $F_i(p)$ onto a small $S^1$. 

Note that although this bundle is defined only up to rotations of $S^1$, 
its fibers are well-defined. 
  
Since $\G_{i+1}$ is normal in $\G$, 
the choice of the covering $\bar{F}_i(p)$ of $F_i(p)$ 
is unambiguous for all $p\in N_i$.  
By using a partition of unity on $N_i$ we can 
glue the forms $\omega_p$ into a global 1-form on $M$ 
which  satisfies the following properties:
\begin{enumerate}[a)]
\item $\omega|_{F(p)}$ is closed  and integral for any $p$;
\item  $\omega|_{F(p)}$ is non-degenerate.
\end{enumerate}
 

 
Integrating $\omega$ over the various $F(p)$'s 
we construct a continuous family of bundles $F_p\to S^1$. 
The level sets
partition each $F(p)$ and hence the whole $M$ into fibers of a fiber bundle,
whose quotient space is then a circle bundle $N_{i+1}$ over $N_i$.


\subsection{}
This gives a good idea of the proof. 
However, to make it precise some extra work has to be done.
In particular,  one has to be careful with the construction of $\omega$. 
To make this construction possible 
we have to keep track of how $F(p)$  was obtained. 
Namely, we have to use that the fiber $F(p)$ 
was obtained by a construction as in Yamaguchi's 
fibration theorem (see~\cite{Yam} or~\cite{BGP}).
This makes the induction proof quite technical.

We now proceed with the rigorous proof of Theorem~\ref{thm:fib}.

\subsection{Proof of Theorem~\ref{thm:fib}} 
Let us denote by  $\tilde M_{n,i}$ 
the Riemannian covering of $M_n$ with respect to $\G_i$. 

For any choice of marked points $p_n$
we have that
$(\tilde M_{n,i},p_n,\pi_1(M))\to (\RR^i,0,\RR^i)$ 
in equivariant Gromov-Hausdorff convergence, 
where $\RR^i$ acts on itself by translations.
Indeed, the limit space must be a 
nonnegatively curved simply connected Alexandrov space, 
and since diam$M_n\to0$ we have that it possesses
a transitive group action by a nilpotent group.  
Then Euclidean space, acting as a group of translations, 
is here the only choice, 
and it is easy to see that the dimension of the limit  must be equal to $i$.

Therefore $(\tilde M_{n},p_n,\pi_1(M))\to (\RR^l,0,\RR^l)$,
and we may also assume that for each $i$ 
we have that $(\tilde M_{n},p_n,\G_i)\to (\RR^l,0, \RR^{l-i})$, 
where $\RR^{l-i}$ is the coordinate subspace of $\RR^l$ 
which corresponds to the first $l-i$ elements of the standard basis. 

Now, let us give a technical definition:


If $R$ is a Riemannian manifold, 
let us denote by $\widetilde\dist_p$ the average of a distance function 
over a small ball around $p$. 
This enables us 
to work with the $C1$  function $\widetilde\dist_p$ 
instead of the Lipschitz function $\dist_p$.

\begin{defn}\label{def:close} 
Let $R_n\to R$ be a sequence of Riemannian $m$-manifolds 
with curvature $\ge \kappa$ which Gromov-Hausdorff converges 
to a Riemannian $m'$-manifold $R$, where $m'\le m$.
A sequence of forms $\omega_n$ on $R_n$ is said to 
$\eps$-approximate a form $\omega$ on $R$, 
if 

(i) 
for any point $p\in R$ there is a neighborhood $U\ni p$ 
which admits a distance chart $f:U\to \R^{m'}$, 
$$f(x)=(\dist_{a_1}(x),\dist_{a_2}(x),...,\dist_{a_{m'}}(x))$$ 
which is a smooth regular map, 
and  

(ii) 
smooth lifts of $f$ to $U_n\i R_n$ 
give, for $n$ large enough, regular maps
$$f_n(x)=(\widetilde\dist_{a_{1,n}}(x),\widetilde\dist_{a_{2,n}}(x),...,
\widetilde\dist_{a_{m',n}}(x))$$
with $a_{i,n}\in M_n$, $a_{i,n}\to a_{n}$
such that 
$$|(f_n\circ f^{-1})^*(\omega)-\omega_n|_{C0}<\eps$$ 
for all sufficiently large $n$. 


\end{defn}

Theorem~\ref{thm:fib} now easily follows from the following lemma:

\begin{lem} 
Given $\eps>0$ there is a sequence of one-forms 
$\{\omega_{1,n},\omega_{2,n},\cdots,\omega_{k,n}\}$ on $\tilde M_n$
with the following properties:
\begin{enumerate}[(i)]
\item For each $i$, $\omega_{i,n}$ is a $\pi_1(M)$-invariant form on $\tilde M_n$.
\item The forms $\omega_{i,n}$ $\eps$-approximate 
the coordinate forms $dx_i$ on $\RR^k$.
In particular, the forms $\{\omega_{i,n}\}$ are nowhere zero 
and almost orthonormal at each point.
\item 
If for any $j<i$ it holds that 
$\omega_{j,n}(X)=\omega_{j,n}(Y)=0$, then $d\omega_{i,n}(X,Y)=0$. 
In particular, 
for each $i$ and all sufficiently large $n$, 
the distribution corresponding to the system of equations 
$$\omega_{j,n}(X)=0 \ \ \text{for all}\ \  j\le i$$ 
defines  on $\tilde M_n$ a foliation $\mathcal F_{i,n}$. 

\item 
If $\tilde F_{i,n}(x)\i \tilde M_n$ denotes 
the fiber of the foliation $\mathcal F_{i,n}$ through the point $x\in \tilde M_n$,
then each $\tilde F_{i,n}(x)$ is $\Gamma_i$-invariant, 
i.e.,  for any $\gamma\in \Gamma_i$
one has that $\tilde F_{i,n}(x)=\tilde F_{i,n}(\gamma x)$.
 Moreover, the action of $\Gamma_i$ on $\tilde F_{i,n}(x)$ is cocompact 
for each $i$. In particular, $\mathcal F_{i,n}$ induces on $M_n$
the structure of a fiber bundle.
\end{enumerate}
\end{lem}


\begin{proof}
We will construct these forms by induction. 
Assume that we have already constructed one-forms 
$\omega_1,\omega_2,\ldots, \omega_{i-1}$ 
which meet all the required properties. 
They give a $\pi_1(M)$-invariant fibration of $\tilde M_n$ by submanifolds 
$\tilde F_{i-1,n}(x)$ through each point $x\in \tilde M_n$, with 
tangent spaces defined by the equations $\omega_j(X)=0$ for $j=1,\ldots,i-1$.

Denote by $\theta:\RR\to [0,1]$ 
a smooth monotone function 
which is equal to $1$ before $0$ and $0$ after $1$. 
Choose numbers $\delta_n>0$ slowly converging to $0$, 
and let $\Theta_{i,n}:\tilde M_n\to \RR_+$ be the function defined by
$$\Theta_{i,n}(x)=\min_{y\in F_{i-1,n}(x)}\theta(|p_n y|/\delta_n).$$ 
Clearly $\Theta_{i,n}$ is a continuous $\G_{i-1}$-invariant function 
which is constant on each $F_{i-1,n}(x)$. 
Moreover, for large $n$,  
$\Theta_{i,n}$ has support in  some $C_i\delta_n$-neighborhood 
of $F_{i-1,n}(p_n)$, and is equal to $1$ 
in some $c_i\delta_n$-neighborhood of $F_{i-1,n}(p_n)$.

Now let $\phi:\RR\to [0,1]$ 
be a smooth nondecreasing function 
which is $0$ before $1/2$ and $1$ after $3/2$.
Consider the form 
$$\omega_{i,n}'=\Theta_{i,n}\cdot d(\phi\circ \widetilde\dist_{\G_{i}a_{i,n}}),$$ 
where $a_{i,n}\in \tilde M_n$ is a sequence of points 
converging to $ -e_i\in \RR^l$,
and $\widetilde\dist_{\G_{i}a_{i,n}}$ is the 
average of $\dist_{\G_{i}x}$ for $x$ in a small ball around $a_{i,n}$.
The support of $\omega_i'$ has two components, 
one which  contains $ p_n$ (notice here that $p_n\to 0\in \R^l$), 
and another which does not. 
(It follows from the construction 
that  the limit of $F_{i-1,n}(p_n)$ is a coordinate plane  in $\RR^l$).

Set $\omega_{i,n}'':=\omega_{i,n}'$ on the component of $ p_n$, 
and let this form be $0$ otherwise.
Clearly, $\omega_{i,n}''$ is then a continuous $\G_{i}$-invariant form 
whose restriction to ${\tilde F_{i-1,n}(x)}$ is exact. 
Moreover, each level set of its integral over 
${\tilde F_{i-1,n}(x)}$ is $\G_{i}$-invariant.  

By construction, the form $\omega''/|\omega''|$ is now
 (in the sense of definition~\ref{def:close})
close to $dx_i$  at the points where 
$|\omega''|\not=0$. 
Take
$$\omega_{i,n}=c\sum_{\gamma\in \G/\G_i} \gamma\omega',$$ 
where the coefficient $c$ is chosen in such a way that  $|\omega_{i,n}(p_n)|=1$.
As $\delta_n$ is a sequence slowly converging to zero, 
we may assume that $\diam (M_n)/\delta_n\to 0$. 
Therefore, $\omega_{i,n}$ is the form we need.
\end{proof}


Notice that the proof of Theorem~5.1 actually also shows 
that the fibers in Theorem ~5.1 are almost nonnegatively curved manifolds
in the generalized sense with $k=l$. Therefore,
 the proof of Theorem~$C$ is complete. \hfill$\qed$



\section{Open questions}
We would like to conclude this work by posing a number of related open questions.
%by describing some possible approaches to attack them.
\begin{quest}
Is it true  that  manifolds which are almost nonnegatively curved in the generalized sense 
are almost nonnegatively curved?
\end{quest}


In view of Theorems ~\ref{intro:main1} and ~\ref{intro:main2} 
it is reasonable to ask the following questions:

\begin{quest}
Is it true that almost nonnegatively curved $m$-manifolds $M^m$
are $C(m)$-nilpotent spaces? 
\end{quest}

It is clear from the proof of Theorems ~\ref{intro:main1} and ~\ref{intro:main2} 
that this is true if the universal cover
of $M^m$ has torsion free integral cohomology.

In view of Theorem~\ref{intro:main2} it is natural to ask the following question:
\begin{quest}
Can one give an explicit bound on $C(m)$ in Theorem ~\ref{intro:main2} ?
\end{quest}


Theorem ~\ref{intro:main1} 
also gives rise to the following question.

\begin{quest}
Do almost nonnegatively curved manifolds admit Riemannian metrics with zero topological entropy ?
\end{quest}

We take interest in this question since it has been shown in \cite{PaPe}
that for the pointed loop space $\Omega M$ of a closed nilpotent manifold $M$ 
which admits a Riemannian metric with zero topological entropy,
$\pi_*(\Omega M)\otimes\QQ$ is finite dimensional.
Notice that for simply connected manifolds this last condition
is equivalent to saying that $M$ is a rationally elliptic space.
Moreover, to include the case of infinite fundamental groups
Totaro has proposed a general definition of an elliptic space
as follows:
A topological space $X$ is {\it elliptic}
if it is homotopy equivalent to a finite CW complex,
it has a finite covering which is a nilpotent space
and the loop space homology of the universal covering of $X$ grows
polynomially with any field of coefficients. 
If the above question has a positive answer,
Theorem ~\ref{intro:main1} and the above result from \cite{PaPe} 
will show that almost nonnegatively curved manifolds
are rationally elliptic in this broader sense.


As was pointed out in the discussion in the Introduction before Theorem~\ref{intro:main3},  it already follows from Yamaguchi's fibration theorem 
and \cite{FY}
that a finite cover of an almost nonnegatively curved manifold 
admits a Serre fibration onto a nilmanifold with simply connected fibers. 
While this is formally weaker than the statement of Theorem~\ref{intro:main3}, it would be interesting to 
have an answer to the following, purely topological, question:


\begin{quest}
Let $F \to M \overset{f}{\to} N$ be a Serre fibration of closed manifolds 
where $N$ is a nilmanifold and $F$ is  simply connected.
Is it true that after passing to a finite cover, the map
$f$ becomes homotopic to a fiber bundle projection ?
\end{quest}























































\small
\bibliographystyle{alpha}
%\bibliography{kpt}

\begin{thebibliography}{HMR75}

\bibitem[And92]{An}
M.~T. Anderson.
\newblock {Hausdorff perturbations of Ricci-flat manifolds and the splitting
  theorem.}
\newblock {\em Duke Math. J.}, 68(1):67--82, 1992.

\bibitem[BGP92]{BGP}
Y.~Burago, M.~Gromov, and G.~Perelman.
\newblock $\text{A.~D. ~Alexandrov's}$ spaces with curvatures bounded from
  below \rm{I}.
\newblock {\em Uspechi Mat. Nauk}, 47:3--51, 1992.

\bibitem[CFG92]{CFG}
J.~Cheeger, K.~Fukaya, and M.~Gromov.
\newblock Nilpotent structures and invariant metrics on collapsed manifolds.
\newblock {\em J. Amer. Math. Soc.}, 5(2):327--372, 1992.

\bibitem[FY92]{FY}
K.~Fukaya and T.~Yamaguchi.
\newblock The fundamental groups of almost nonnegatively curved manifolds.
\newblock {\em Ann. of Math. (2)}, 136(2):253--333, 1992.

\bibitem[Gal83]{Ga}
S.~Gallot.
\newblock In\'egalit\'es isop\'erim\'etriques, courbure de {R}icci et
  invariants g\'eom\'etriques. {II}.
\newblock {\em C. R. Acad. Sci. Paris S\'er. I Math.}, 296(8):365--368, 1983.

\bibitem[Gro78]{G1}
M.~Gromov.
\newblock {Almost flat manifolds.}
\newblock {\em J. Differ. Geom.}, 13:231--241, 1978.

\bibitem[Gro80]{G4}
M.~Gromov.
\newblock Synthetic geometry in {R}iemannian manifolds.
\newblock In {\em Proceedings of the International Congress of Mathematicians
  (Helsinki, 1978)}, pages 415--419, Helsinki, 1980. Acad. Sci. Fennica.

\bibitem[Gro81]{G2}
M.~Gromov.
\newblock {Curvature, diameter and Betti numbers.}
\newblock {\em Comment. Math. Helv.}, 56:179--195, 1981.

\bibitem[Gro96]{G3}
M.~Gromov.
\newblock Positive curvature, macroscopic dimension, spectral gaps and higher
  signatures.
\newblock In {\em Functional analysis on the eve of the 21st century, Vol.\ II
  (New Brunswick, NJ, 1993)}, pages 1--213. Birkh\"auser Boston, 1996.

\bibitem[HMR75]{HMR}
P.~Hilton, G.~Mislin, and J.~Roitberg.
\newblock {\em Localization of nilpotent groups and spaces}.
\newblock North-Holland Publishing Co., Amsterdam, 1975.
\newblock North-Holland Mathematics Studies, No. 15, Notas de Matem\'atica, No.
  55. [Notes on Mathematics, No. 55].

\bibitem[LeB01]{Lb}
C.~LeBrun.
\newblock Ricci curvature, minimal volumes, and {S}eiberg-{W}itten theory.
\newblock {\em Invent. Math.}, 145(2):279--316, 2001.

\bibitem[Per91]{Per}
G.~Perelman.
\newblock Alexandrov spaces with curvatures bounded from below \rm{II}.
\newblock preprint, 1991.

\bibitem[Pet95]{Ptr2}
A.~Petrunin.
\newblock {\em Quasigeodesics in multidimensional Alexandrov spaces}.
\newblock PhD thesis, University of Illinois at Urbana Champaign, 1995.

\bibitem[Pet98]{Ptr3}
A.~Petrunin.
\newblock {Parallel transportation for Alexandrov space with curvature bounded
  below.}
\newblock {\em Geom. Funct. Anal.}, 8(1):123--148, 1998.

\bibitem[PP96]{PP}
G.~Perelman and A.~Petrunin.
\newblock Quasigeodesics and gradient curves in alexandrov spaces.
\newblock preprint, http://www.math.psu.edu/petrunin/papers/papers.html, 1996.

\bibitem[PP04]{PaPe}
G.~Paternain and J.~Petean.
\newblock Zero entropy and bounded topology.
\newblock preprint math.DG/0406051, 2004.

\bibitem[Sch89]{Sch}
R.~M. Schoen.
\newblock Variational theory for the total scalar curvature functional for
  {R}iemannian metrics and related topics.
\newblock In {\em Topics in calculus of variations (Montecatini Terme, 1987)},
  volume 1365 of {\em Lecture Notes in Math.}, pages 120--154. Springer,
  Berlin, 1989.

\bibitem[Sha78]{Shar}
V.A. Sharafutdinov.
\newblock {The Pogorelov - Klingenberg theorem for manifolds homeomorphic to
  $\RR^n$.}
\newblock {\em Sib. Math. J.}, 18:649--657, 1978.

\bibitem[ST04]{ST}
L.~J. Schwachhoefer and W.~Tuschmann.
\newblock {Meterics of positive Ricci curvature on quotient spaces.}
\newblock {\em Math. Ann.}, 330(1):59--91, 2004.

\bibitem[Yam91]{Yam}
T.~Yamaguchi.
\newblock Collapsing and pinching under a lower curvature bound.
\newblock {\em Ann. of Math.}, 133:317--357, 1991.

\end{thebibliography}

\end{document}

math0506273.extracted_bib
\begin{thebibliography}{HMR75}

\bibitem[And92]{An}
M.~T. Anderson.
\newblock {Hausdorff perturbations of Ricci-flat manifolds and the splitting
  theorem.}
\newblock {\em Duke Math. J.}, 68(1):67--82, 1992.

\bibitem[BGP92]{BGP}
Y.~Burago, M.~Gromov, and G.~Perelman.
\newblock $\text{A.~D. ~Alexandrov's}$ spaces with curvatures bounded from
  below \rm{I}.
\newblock {\em Uspechi Mat. Nauk}, 47:3--51, 1992.

\bibitem[CFG92]{CFG}
J.~Cheeger, K.~Fukaya, and M.~Gromov.
\newblock Nilpotent structures and invariant metrics on collapsed manifolds.
\newblock {\em J. Amer. Math. Soc.}, 5(2):327--372, 1992.

\bibitem[FY92]{FY}
K.~Fukaya and T.~Yamaguchi.
\newblock The fundamental groups of almost nonnegatively curved manifolds.
\newblock {\em Ann. of Math. (2)}, 136(2):253--333, 1992.

\bibitem[Gal83]{Ga}
S.~Gallot.
\newblock In\'egalit\'es isop\'erim\'etriques, courbure de {R}icci et
  invariants g\'eom\'etriques. {II}.
\newblock {\em C. R. Acad. Sci. Paris S\'er. I Math.}, 296(8):365--368, 1983.

\bibitem[Gro78]{G1}
M.~Gromov.
\newblock {Almost flat manifolds.}
\newblock {\em J. Differ. Geom.}, 13:231--241, 1978.

\bibitem[Gro80]{G4}
M.~Gromov.
\newblock Synthetic geometry in {R}iemannian manifolds.
\newblock In {\em Proceedings of the International Congress of Mathematicians
  (Helsinki, 1978)}, pages 415--419, Helsinki, 1980. Acad. Sci. Fennica.

\bibitem[Gro81]{G2}
M.~Gromov.
\newblock {Curvature, diameter and Betti numbers.}
\newblock {\em Comment. Math. Helv.}, 56:179--195, 1981.

\bibitem[Gro96]{G3}
M.~Gromov.
\newblock Positive curvature, macroscopic dimension, spectral gaps and higher
  signatures.
\newblock In {\em Functional analysis on the eve of the 21st century, Vol.\ II
  (New Brunswick, NJ, 1993)}, pages 1--213. Birkh\"auser Boston, 1996.

\bibitem[HMR75]{HMR}
P.~Hilton, G.~Mislin, and J.~Roitberg.
\newblock {\em Localization of nilpotent groups and spaces}.
\newblock North-Holland Publishing Co., Amsterdam, 1975.
\newblock North-Holland Mathematics Studies, No. 15, Notas de Matem\'atica, No.
  55. [Notes on Mathematics, No. 55].

\bibitem[LeB01]{Lb}
C.~LeBrun.
\newblock Ricci curvature, minimal volumes, and {S}eiberg-{W}itten theory.
\newblock {\em Invent. Math.}, 145(2):279--316, 2001.

\bibitem[Per91]{Per}
G.~Perelman.
\newblock Alexandrov spaces with curvatures bounded from below \rm{II}.
\newblock preprint, 1991.

\bibitem[Pet95]{Ptr2}
A.~Petrunin.
\newblock {\em Quasigeodesics in multidimensional Alexandrov spaces}.
\newblock PhD thesis, University of Illinois at Urbana Champaign, 1995.

\bibitem[Pet98]{Ptr3}
A.~Petrunin.
\newblock {Parallel transportation for Alexandrov space with curvature bounded
  below.}
\newblock {\em Geom. Funct. Anal.}, 8(1):123--148, 1998.

\bibitem[PP96]{PP}
G.~Perelman and A.~Petrunin.
\newblock Quasigeodesics and gradient curves in alexandrov spaces.
\newblock preprint, http://www.math.psu.edu/petrunin/papers/papers.html, 1996.

\bibitem[PP04]{PaPe}
G.~Paternain and J.~Petean.
\newblock Zero entropy and bounded topology.
\newblock preprint math.DG/0406051, 2004.

\bibitem[Sch89]{Sch}
R.~M. Schoen.
\newblock Variational theory for the total scalar curvature functional for
  {R}iemannian metrics and related topics.
\newblock In {\em Topics in calculus of variations (Montecatini Terme, 1987)},
  volume 1365 of {\em Lecture Notes in Math.}, pages 120--154. Springer,
  Berlin, 1989.

\bibitem[Sha78]{Shar}
V.A. Sharafutdinov.
\newblock {The Pogorelov - Klingenberg theorem for manifolds homeomorphic to
  $\RR^n$.}
\newblock {\em Sib. Math. J.}, 18:649--657, 1978.

\bibitem[ST04]{ST}
L.~J. Schwachhoefer and W.~Tuschmann.
\newblock {Meterics of positive Ricci curvature on quotient spaces.}
\newblock {\em Math. Ann.}, 330(1):59--91, 2004.

\bibitem[Yam91]{Yam}
T.~Yamaguchi.
\newblock Collapsing and pinching under a lower curvature bound.
\newblock {\em Ann. of Math.}, 133:317--357, 1991.

\end{thebibliography}