Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Volume rigidity at ideal points of the character variety of hyperbolic
  3-manifolds   Given the fundamental group $\Gamma$ of a finite-volume complete hyperbolic
$3$-manifold $M$, it is possible to associate to any representation
$\rho:\Gamma \rightarrow \text{Isom}(\mathbb{H}^3)$ a numerical invariant
called volume. This invariant is bounded by the hyperbolic volume of $M$ and
satisfies a rigidity condition: if the volume of $\rho$ is maximal, then $\rho$
must be conjugated to the holonomy of the hyperbolic structure of $M$. This
paper generalizes this rigidity result by showing that if a sequence of
representations of $\Gamma$ into $\text{Isom}(\mathbb{H}^3)$ satisfies $\lim_{n
\to \infty} \text{Vol}(\rho_n) = \text{Vol}(M)$, then there must exist a
sequence of elements $g_n \in \text{Isom}(\mathbb{H}^3)$ such that the
representations $g_n \circ \rho_n \circ g_n^{-1}$ converge to the holonomy of
$M$. In particular if the sequence $\rho_n$ converges to an ideal point of the
character variety, then the sequence of volumes must stay away from the
maximum. We conclude by generalizing the result to the case of $k$-manifolds
and representations in $\text{Isom}(\mathbb H^m)$, where $m\geq k$.

18
The categories of this paper are: econ
33
The categories of this paper are: plasm-ph
13
The categories of this paper are: quant-ph
37
The categories of this paper are: supr-con
21
The categories of this paper are: adap-org
30
The categories of this paper are: q-alg
9
The categories of this paper are: nlin
25
The categories of this paper are: patt-sol
28
The categories of this paper are: alg-geom
23
The categories of this paper are: solv-int
35
The categories of this paper are: mtrl-th
31
The categories of this paper are: ao-sci
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The categories of this paper are: bayes-an
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The categories of this paper are: eess
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The categories of this paper are: math-ph
4
The categories of this paper are: cond-mat