Some finiteness results for groups with bounded algebraic entropy. (English. Abridged French version) Zbl 1064.20038
Summary: We prove the following fact: the number of elements of any generating set \(S\) of a discrete group \(G\) is bounded from above if we assume that the algebraic entropy of \(G\) with respect to \(S\) is smaller than some universal constant and the existence of a finite index subgroup of \(G\) with some hyperbolicity properties. We deduce some finiteness results for the pairs \((G,S)\) when there exists a system of relations of (universally) bounded length, as it is the case for word hyperbolic groups or fundamental groups of manifolds. In this last case, the results are of geometric interest.
MSC:
| 20F05 | Generators, relations, and presentations of groups |
| 20F67 | Hyperbolic groups and nonpositively curved groups |
| 57M05 | Fundamental group, presentations, free differential calculus |
Keywords:
numbers of elements; generating sets; algebraic entropy; subgroups of finite index; word hyperbolic groups; fundamental groupsReferences:
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