Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A rank rigidity result for CAT(0) spaces with one-dimensional Tits
  boundaries   We prove the following rank rigidity result for proper CAT(0) spaces with
one-dimensional Tits boundaries: Let $\Gamma$ be a group acting properly
discontinuously, cocompactly, and by isometries on such a space $X$. If the
Tits diameter of $\partial X$ equals $\pi$ and $\Gamma$ does not act minimally
on $\partial X$, then $\partial X$ is a spherical building or a spherical join.
If $X$ is also geodesically complete, then $X$ is a Euclidean building, higher
rank symmetric space, or a nontrivial product. Much of the proof, which
involves finding a Tits-closed convex building-like subset of $\partial X$,
does not require the Tits diameter to be $\pi$, and we give an alternate
condition that guarantees rigidity when this hypothesis is removed, which is
that a certain invariant of the group action be even.

8
The categories of this paper are: hep-ex
34
The categories of this paper are: chem-ph
5
The categories of this paper are: gr-qc
7
The categories of this paper are: hep-th
20
The categories of this paper are: acc-phys
12
The categories of this paper are: nucl-th
3
The categories of this paper are: physics
22
The categories of this paper are: chao-dyn
6
The categories of this paper are: astro-ph
30
The categories of this paper are: q-alg
0
The categories of this paper are: hep-ph
26
The categories of this paper are: bayes-an
9
The categories of this paper are: nlin
14
The categories of this paper are: math-ph
24
The categories of this paper are: dg-ga
11
The categories of this paper are: hep-lat