The $R_\infty$ property for nilpotent quotients of surface groups

TL;DR

This paper determines the minimal levels of nilpotent and solvable quotients of surface groups that retain the $R_{ abla}$ property, revealing specific degrees for orientable and non-orientable surfaces.

Contribution

It introduces the concepts of $R_{ abla}$-nilpotency and $R_{ abla}$-solvability degrees for surface groups and computes their exact values for various cases.

Findings

$R_{ abla}$-nilpotency degree is 4 for orientable surfaces of genus > 1.

$R_{ abla}$-nilpotency degree is 2(g-1) for non-orientable surfaces with genus g > 2.

$R_{ abla}$-solvability degree of orientable surface groups is 2.

Abstract

It is well known that when $G$ is the fundamental group of a closed surface of negative Euler characteristic, it has the $R_{\infty}$ property. In this work we compute the least integer $c$, {\it called the $R_{\infty}$-nilpotency degree of $G$}, such that the group $G/ \gamma_{c+1}(G)$ has the $R_{\infty}$ property, where $\gamma_r(G)$ is the $r$-th term of the lower central series of $G$. We show that $c=4$ for $G$ the fundamental group of any orientable closed surface $S_g$ of genus $g>1$. For the fundamental group of the non-orientable surface $N_g$ (the connected sum of $g$ projective planes) this number is $2(g-1)$ (when $g>2$). A similar concept is introduced using the derived series $G^{(r)}$ of a group $G$. Namely {\it the $R_{\infty}$-solvability degree of $G$}, which is the least integer $c$ such that the group $G/G^{(c)}$ has the $R_{\infty}$ property. We show that the fundamental group of an orientable closed surface $S_g$ has $R_{\infty}$-solvability degree $2$.