Rational growth in virtually abelian groups

Abstract

We initiate the study of the twisted conjugacy growth series of a finitely generated group, the formal power series associated to the twisted conjugacy growth function. Our main result is that this series is always an N-rational function when the group in question is virtually abelian, and that this function can be explicitly computed from appropriate information about the group and the twisting endomorphism. We also show that the relative growth series of any twisted conjugacy class in a finitely generated virtually abelian group is itself an explicitly computable N-rational function.

Formal conjugacy growth in graph products I

“…In combination with inequalities ( 7), ( 8), (9), and (10) above, Corollary 4.5 shows that RC(σ V ) ≤ RC(σ V ).…”

“…The first conjugacy growth series computations appeared in the work of Rivin ([18,19]) on free groups, and it is striking that, even for free groups with standard generating sets, the series are transcendental, and their formulas rather complicated. More generally and systematically, conjugacy growth series and languages featured in [2,6,7,3,9,17], where virtually abelian groups, acylindrically hyperbolic groups, free and wreath products, and more, were explored.…”

Conjugacy growth in the higher Heisenberg groups

“…Ciobanu, Hermiller, Holt and Rees [7] and Antolín and Ciobanu [1] confirmed a conjecture of Rivin [29], that the conjugacy growth series of a hyperbolic group is a rational function if and only if the group is virtually cyclic. Continuing in a similar vein, the author proved the rationality of the series for all virtually abelian groups [12], and Gekhtman and Yang proved transcendence for relatively hyperbolic groups and certain acylindrically hyperbolic groups [17]. All of the above results apply to any choice of finite generating set, and lead to the following conjecture: Conjecture 1.1 (Conjecture 7.2 of [6]).…”

Equations in virtually abelian groups: languages and growth

“…The following definitions and results follow Benson's work [Ben83], where it is proved that virtually abelian groups have rational (standard) growth series. More recently, polyhedral sets have again been used to prove rationality of various growth series of groups ([DS19], [Eve19]). Definition 2.2.…”

“…This relative growth has been studied in many papers, including [DO15]. The relative growth series of any subgroup of virtually abelian groups was shown to be rational in [Eve19]. In Section 4 we consider the relative growth of the solution set of any system of equations in a virtually abelian group, either as a subset of the group, or as a set of tuples of group elements (with an appropriate metric) in the case of equations in multiple variables.…”