Rational Growth in Virtually Abelian Groups

Abstract

We show that any subgroup of a finitely generated virtually abelian group G grows rationally relative to G, that the set of right cosets of any subgroup of G grows rationally, and that the set of conjugacy classes of G grows rationally. These results hold regardless of the choice of finite weighted generating set for G.

The conjugacy growth of the soluble Baumslag-Solitar groups

“…This paper provides further confirmation for the conjecture (see [11]) that the only groups with rational conjugacy growth series are the virtually abelian ones. It also provides further confirmation for the conjecture that the conjugacy and standard growth rates in finitely presented groups are equal; this was already observed for hyperbolic [1], relatively hyperbolic [14], most graph products [7] and lamplighter groups [18].…”

“…Conjecture 26 (see also [11]). The conjugacy growth series of any finitely presented group that is not virtually abelian is transcendental.…”

“…For any n ≥ 0, the conjugacy growth function c G,S (n) of a finitely generated group G, with respect to some finite generating set S, counts the number of conjugacy classes intersecting the ball of radius n in the Cayley graph of G with respect to S. The conjugacy growth series of G with respect to S is then the generating function for the sequence c G,S (n). There are numerous results in the literature about the asymptotics of conjugacy growth [9,10,15,17], as well as about the behaviour of conjugacy growth series [1,5,6,11,18,19], for important classes of groups. Of particular relevance here is the work [2] of Breuillard and Cornulier, who showed that the function c G,S (n) grows exponentially for finitely generated soluble groups that are not virtually nilpotent, such as the soluble Baumslag-Solitar groups BS(1, k) = ⟨a, t tat −1 = a k ⟩, k ≥ 2.…”