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In mathematics, the free group FS over a given set S consists of all words that can be built from members of S, considering two words to be different unless their equality follows from the group axioms (e.g. st = suu−1t, but s ≠ t−1 for s,t,u ∈ S). The members of S are called generators of FS, and the number of generators is the rank of the free group. An arbitrary group G is called free if it is isomorphic to FS for some subset S of G, that is, if there is a subset S of G such that every element of G can be written in exactly one way as a product of finitely many elements of S and their inverses (disregarding trivial variations such as st = suu−1t). A related but different notion is a free abelian group; both notions are particular instances of a free object from universal algebra. As such, free groups are defined by their universal property. History Free groups first arose in the study of hyperbolic geometry, as examples of Fuchsian groups (discrete groups acting by is

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For a group G generated by k elements, the Nielsen equivalence classes are defined as orbits of the action of AutF(k), the automorphism group of the free group of rank k, on the set of generating k-tuples of G. Let p >= 3 be prime and G(p) the Gupta-Sidki p-group. We prove that there are infinitely many Nielsen equivalence classes on generating pairs of G(p).

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Given a group Gamma, we establish a connection between the unitarisability of its uniformly bounded representations and the asymptotic behaviour of the isoperimetric constants of Cayley graphs of Gamma for increasingly large generating sets. The connection hinges on an analytic invariant Lit(Gamma) is an element of [0, infinity] which we call the Littlewood exponent. Finiteness, amenability, unitarisability and the existence of free subgroups are related respectively to the thresholds 0, 1, 2 and infinity for Lit(Gamma). Using graphical small cancellation theory, we prove that there exist groups Gamma for which 1 < Lit(Gamma) < infinity. Further applications, examples and problems are discussed. (C) 2020 Elsevier Inc. All rights reserved.

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