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Some finiteness results for groups with bounded algebraic entropy. (English) Zbl 1221.20030

Let \(G\) be a finitely generated group and \(S\) a symmetric finite generating set of \(G\), endowing \(G\) with a word metric. For any positive integer \(n\), \(f(n)\) denotes the cardinality of the closed ball of radius \(n\) centered at the identity. The entropy \(\mathrm{Ent}_S(G)\) is then defined by \[ \mathrm{Ent}_S(G)=\lim_{n\to\infty}\frac{\log f(n)}{n}. \] In the paper under review, the author establishes an upper bound on the cardinality of \(S\) when the entropy \(\mathrm{Ent}_S(G)\) is bounded from above by some universal constant, under the assumption that \(G\) contains a finite-index subgroup sharing some algebraic properties with torsion-free word-hyperbolic groups, which he calls an \(N\)-Abelian group.
As an application, the author obtains some finiteness results on the class of word-hyperbolic groups and on the class of fundamental groups of Riemannian manifolds.

MSC:

20F65 Geometric group theory
20F67 Hyperbolic groups and nonpositively curved groups
53C20 Global Riemannian geometry, including pinching
20F05 Generators, relations, and presentations of groups
37A35 Entropy and other invariants, isomorphism, classification in ergodic theory
28D20 Entropy and other invariants
57M05 Fundamental group, presentations, free differential calculus
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