The Geometric Invariants of Group Extensions Part I: Finite Extensions

TL;DR

This paper investigates the geometric invariants of finite group extensions, computes the {\

Contribution

It provides explicit calculations of the {\

Findings

Computed {\

Constructed a group F semidirect Z_2 with the R-infinity property.

Built a finite extension G with a finitely generated G' but an infinitely generated H' in a finite index subgroup.

Abstract

In this note, we compute the {\Sigma}^1(G) invariant when 1 {\to} H {\to} G {\to} K {\to} 1 is a short exact sequence of finitely generated groups with K finite. As an application, we construct a group F semidirect Z_2 where F is the R. Thompson's group F and show that F semidirect Z_2 has the R-infinity property while F is not characteristic. Furthermore, we construct a finite extension G with finitely generated commutator subgroup G' but has a finite index normal subgroup H with infinitely generated H'.