Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Entropy and finiteness of groups with acylindrical splittings   We prove that there exists a positive, explicit function $F(k, E)$ such that,
for any group $G$ admitting a $k$-acylindrical splitting and any generating set
$S$ of $G$ with $\mathrm{Ent}(G,S)<E$, we have $|S| \leq F(k, E)$. We deduce
corresponding finiteness results for classes of groups possessing acylindrical
splittings and acting geometrically with bounded entropy: for instance,
$D$-quasiconvex $k$-malnormal amalgamated products acting on
$\delta$-hyperbolic spaces or on $CAT(0)$-spaces with entropy bounded by $E$. A
number of finiteness results for interesting families of Riemannian or metric
spaces with bounded entropy and diameter also follow: Riemannian 2-orbifolds,
non-geometric $3$-manifolds, higher dimensional graph manifolds and
cusp-decomposable manifolds, ramified coverings and, more generally,
CAT(0)-groups with negatively curved splittings.

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The categories of this paper are: hep-lat
18
The categories of this paper are: econ
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The categories of this paper are: math-ph
37
The categories of this paper are: supr-con
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The categories of this paper are: nucl-th
3
The categories of this paper are: physics
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The categories of this paper are: dg-ga
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The categories of this paper are: acc-phys
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The categories of this paper are: stat
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The categories of this paper are: adap-org
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The categories of this paper are: eess
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The categories of this paper are: bayes-an
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The categories of this paper are: cmp-lg
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The categories of this paper are: comp-gas
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The categories of this paper are: quant-ph
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The categories of this paper are: funct-an