Torsion-free word hyperbolic groups are noncommutatively slender

In this paper, we prove the claim given in the title. A group [Formula: see text] is noncommutatively slender if each map from the fundamental group of the Hawaiian Earring to [Formula: see text] factors through projection to a canonical free subgroup. Higman, in his seminal 1952 paper [Unrestricted free products and varieties of topological groups, J. London Math. Soc. 27 (1952) 73–81], proved that free groups are noncommutatively slender. Such groups were first defined by Eda in [Free [Formula: see text]-products and noncommutatively slender groups, J. Algebra 148 (1992) 243–263]. Eda has asked which finitely presented groups are noncommutatively slender. This result demonstrates that random finitely presented groups in the few-relator sense are noncommutatively slender.

Metric Characterizations of Superreflexivity in Terms of Word Hyperbolic Groups and Finite Graphs

We show that superreflexivity can be characterized in terms of bilipschitz embeddability of word hyperbolic groups.We compare characterizations of superrefiexivity in terms of diamond graphs and binary trees.We show that there exist sequences of series-parallel graphs of increasing topological complexitywhich admit uniformly bilipschitz embeddings into a Hilbert space, and thus do not characterize superrefiexivity.

Combable functions, quasimorphisms, and the central limit theorem

A function on a discrete group is weakly combable if its discrete derivative with respect to a combing can be calculated by a finite-state automaton. A weakly combable function is bicombable if it is Lipschitz in both the left- and right-invariant word metrics. Examples of bicombable functions on word-hyperbolic groups include:(1)homomorphisms to ℤ;(2)word length with respect to a finite generating set;(3)most known explicit constructions of quasimorphisms (e.g. the Epstein–Fujiwara counting quasimorphisms).We show that bicombable functions on word-hyperbolic groups satisfy acentral limit theorem: if$\overline {\phi }_n$is the value of ϕ on a random element of word lengthn(in a certain sense), there areEandσfor which there is convergence in the sense of distribution$n^{-1/2}(\overline {\phi }_n - nE) \to N(0,\sigma )$, whereN(0,σ) denotes the normal distribution with standard deviationσ. As a corollary, we show that ifS1andS2are any two finite generating sets forG, there is an algebraic numberλ1,2depending onS1andS2such that almost every word of lengthnin theS1metric has word lengthn⋅λ1,2in theS2metric, with error of size$O(\sqrt {n})$.

Quasiregular self-mappings of manifolds and word hyperbolic groups

An extension of a result of Sela shows that if Γ is a torsion-free word hyperbolic group, then the only homomorphisms Γ→Γ with finite-index image are the automorphisms. It follows from this result and properties of quasiregular mappings, that if M is a closed Riemannian n-manifold with negative sectional curvature ($n\neq 4$), then every quasiregular mapping f:M→M is a homeomorphism. In the constant-curvature case the dimension restriction is not necessary and Mostow rigidity implies that f is homotopic to an isometry. This is to be contrasted with the fact that every such manifold admits a non-homeomorphic light open self-mapping. We present similar results for more general quotients of hyperbolic space and quasiregular mappings between them. For instance, we establish that besides covering projections there are no π1-injective proper quasiregular mappings f:M→N between hyperbolic 3-manifolds M and N with non-elementary fundamental group.

THE LINEARITY OF THE CONJUGACY PROBLEM IN WORD-HYPERBOLIC GROUPS

The main result proved in this paper is that the conjugacy problem in word-hyperbolic groups is solvable in linear time. This is using a standard RAM model of computation, in which basic arithmetical operations on integers are assumed to take place in constant time. The constants involved in the linear time solution are all computable explicitly. We also give a proof of the result of Mike Shapiro that in a word-hyperbolic group a word in the generators can be transformed into short-lex normal form in linear time. This is used in the proof of our main theorem, but is a significant theoretical result of independent interest, which deserves to be in the literature. Previously the best known result was a quadratic estimate.