The Geometric Invariants of Group Extensions

TL;DR

This paper investigates the geometric invariants of group extensions, specifically computing {\Sigma}^n(G) and {\Omega}^n(G) for certain exact sequences, and explores conditions for the R_{\infty} property in such groups.

Contribution

It provides explicit calculations of invariants for group extensions with finite quotients and establishes new criteria for the R_{\infty} property in these groups.

Findings

Computed {\Sigma}^n(G) and {\Omega}^n(G) invariants for specific group extensions.

Identified conditions under which groups have the R_{\infty} property.

Constructed a new example of a group with the R_{\infty} property involving Thompson's group F.

Abstract

In this paper, we compute the {\Sigma}^n(G) and {\Omega}^n(G) invariants when 1 \rightarrow H \rightarrow G \rightarrow K \rightarrow 1 is a short exact sequence of finitely generated groups with K finite. We also give sufficient conditions for G to have the R_{\infty} property in terms of {\Omega}^n(H) and {\Omega}^n(K) when either K is finite or the sequence splits. As an application, we construct a group F \rtimes? Z_2 where F is the R. Thompson's group F and show that F \rtimes Z_2 has the R_{\infty} property while F is not characteristic.