Genericity of contracting elements in groups

Genericity of Filling Elements

An element of a finitely generated non-Abelian free group F (X) is said to be filling if that element has positive translation length in every very small minimal isometric action of F (X) on an R-tree. We give a proof that the set of filling elements of F (X) is exponentially F (X)-generic in the sense of Arzhantseva and Ol’shanskĭı. We also provide an algebraic sufficient condition for an elem...

A Fractal Non-Contracting Class of Automata Groups

Pairwise‎ ‎non-commuting elements in finite metacyclic $2$-groups and some finite $p$-groups

Let $G$ be a finite group‎. ‎A subset $X$ of $G$ is a set of pairwise non-commuting elements‎ ‎if any two distinct elements of $X$ do not commute‎. ‎In this paper‎ ‎we determine the maximum size of these subsets in any finite‎ ‎non-abelian metacyclic $2$-group and in any finite non-abelian $p$-group with an abelian maximal subgroup‎.

a fractal non-contracting class of automata groups

Real Elements in Spin Groups

Let F be a field of characteristic 6= 2. Let G be an algebraic group defined over F . An element t ∈ G(F ) is called real if there exists s ∈ G(F ) such that sts = t. A semisimple element t in GLn(F ), SLn(F ), O(q), SO(q), Sp(2n) and the groups of type G2 over F is real if and only if t = τ1τ2 where τ 2 1 = ±1 = τ 2 2 (ref. [ST1, ST2]). In this paper we extend this result to the semisimple ele...