Small cancellation groups and translation numbers

Abstract

Abstract. In this paper we prove that C(4)-T(4)-P, C(3)-T(6)-P and C(6)-P small cancellation groups are translation discrete in the strongest possible sense and that in these groups for any g and any n there is an algorithm deciding whether or not the equation x n = g has a solution. There is also an algorithm for calculating for each g the maximum n such that g is an n-th power of some element. We also note that these groups cannot contain isomorphic copies of the group of p-adic fractions and so in particular of the group of rational numbers. Besides we show that for C (4) − T (4) and C (3) − T (6) groups all translation numbers are rational and have bounded denominators. IntroductionIn [GS1] S.Gersten and H.Short introduced the notion of translation numbers for finitely generated groups. This concept was inspired by some considerations about groups acting on R-trees and about the action of the fundamental group of a closed riemannian manifold of nonpositive curvature on the universal cover of this manifold (see, for example, the book of W.Ballmann, M.Gromov and V.Schroeder, "Manifolds of nonpositive curvature" [BGrS]).Definition. Let G be a group and let X be a finite generating set of G closed under inversion. Then any element g in G can be expressed as a product g = x 1 ·x 2 ·· · ··x n where x i ∈ X; we call the minimal such n the X-length of g and denote it l X (g).Then for any g ∈ G we define the translation number of g with respect to X as follows:(It is noted in [GS1] that this limit always exists.)From now on we will assume that all generating sets for all groups in this section are closed under inversions. It turns out (see [GS1] for proofs) that if G is a group and X is a finite generating set, then:(a) τ X (g) = τ X (g −1 ) ≤ l X (g); (b) τ X (g) = τ X (hgh −1 ); (c) if g is an element of finite order then τ X (g) = 0; (d) τ X (g n ) = nτ X (g) for any integer n and for any g ∈ G;

Braid groups of imprimitive complex reflection groups

“…When A is a set of group generators, |g| A and t G,A (g) indicate |g| A∪A −1 and t G,A∪A −1 (g), respectively. Kapovich [Kap97] and Conner [Con00] suggested the following notions: a finitely generated group G is said to be (i) translation separable if for some (and hence for any) finite set X of semigroup generators for G the translation numbers of non-torsion elements are strictly positive; (ii) translation discrete if it is translation separable and for some (and hence for any) finite set X of semigroup generators for G the set t G,X (G) has 0 as an isolated point; (iii) strongly translation discrete if it is translation separable and for some (and hence for any) finite set X of semigroup generators for G and for any real number r the number of conjugacy classes [g] = {h −1 gh : h ∈ G} with t G,X (g) ≤ r is finite. (The translation number is constant on each conjugacy class.…”

Negative curvature in graphical small cancellation groups

“…Let | · | be the word length in G(Γ) with respect to S. The translation length of an element g ∈ G(Γ) is: τ (g) := lim n→∞ |g n | n Conner [16] calls a group whose non-torsion elements have translation length bounded away from zero translation discrete. Hyperbolic groups [50], CAT(0) groups [17], and finitely presented groups satisfying various classical small cancellation conditions [34] are translation discrete.…”

A New Proof That Alternating Links Are Non-Trivial

“…The Geodesic Characterization Theorem appeared implicitly in [1] and [7]. A formal proof can be found in [9], where the Geodesic Characterization Theorem appears as Lemma 3.2.…”