The Geometric Invariants of Group Extensions Part II: Split Extensions

TL;DR

This paper computes the ^1(G) invariant for split group extensions and applies it to braid groups on surfaces, with implications for twisted conjugacy and subgroup generation.

Contribution

It provides explicit calculations of the ^1(G) invariant for split extensions and surface braid groups, extending previous theoretical frameworks.

Findings

^1(G) computed for split extensions

^1(G) applied to braid groups on surfaces

Results inform twisted conjugacy and subgroup properties

Abstract

We compute the {\Omega}^1(G) invariant when 1 {\to} H {\to} G {\to} K {\to} 1 is a split short exact sequence. We use this result to compute the invariant for pure and full braid groups on compact surfaces. Applications to twisted conjugacy classes and to finite generation of commutator subgroups are also discussed.