Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A Matsumoto-Mostow result for Zimmer's cocycles of hyperbolic lattices   As for the theory of maximal representations, we introduce the volume of a
Zimmer's cocycle $\Gamma \times X \rightarrow \mbox{PO}^\circ(n, 1)$, where
$\Gamma$ is a torsion-free (non-)uniform lattice in $\mbox{PO}^\circ(n, 1)$,
with $n \geq 3$, and $X$ is a suitable standard Borel probability
$\Gamma$-space. Our numerical invariant extends the volume of representations
for (non-)uniform lattices to measurable cocycles and in the uniform setting it
agrees with the generalized version of the Euler number of self-couplings. We
prove that our volume of cocycles satisfies a Milnor-Wood type inequality in
terms of the volume of the manifold $\Gamma \backslash \mathbb{H}^n$. This
invariant can be interpreted as a suitable multiplicative constant between
bounded cohomology classes. This allows us to characterize maximal cocycles for
being cohomologous to the cocycle induced by the standard lattice embedding via
a measurable map $X \rightarrow \mbox{PO}(n, 1)$ with essentially constant
sign.
  As a by-product of our rigidity result for the volume of cocycles, we give a
new proof of the mapping degree theorem. This allows us to provide a complete
characterization of maps homotopic to local isometries between closed
hyperbolic manifolds in terms of maximal cocycles.
  In dimension $n = 2$, we introduce the notion of Euler number of measurable
cocycles associated to closed surface groups. It extends the classic Euler
number of representations and it agrees with the generalized version of the
Euler number of self-couplings up to a multiplicative constant. We show a
Milnor-Wood type inequality whose upper bound is given by the modulus of the
Euler characteristic. This gives an alternative proof of the same result for
the generalized version of the Euler number of self-couplings. Finally, we
characterize maximal cocycles as those which are cohomologous to the one
induced by a hyperbolization.

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The categories of this paper are: q-bio
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The categories of this paper are: cs
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The categories of this paper are: q-alg
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The categories of this paper are: cond-mat
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The categories of this paper are: nucl-th
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The categories of this paper are: astro-ph
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The categories of this paper are: alg-geom
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The categories of this paper are: ao-sci
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The categories of this paper are: adap-org
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The categories of this paper are: hep-lat
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The categories of this paper are: comp-gas
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The categories of this paper are: hep-ph
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The categories of this paper are: gr-qc
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The categories of this paper are: quant-ph
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The categories of this paper are: nucl-ex
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The categories of this paper are: funct-an