2D problems in groups

TL;DR

This paper explores a conjecture related to group theory, focusing on the stabilization of deficiency in finite index subgroups and its connections to the D2 Problem and Relation Gap problem, with verification of the pro-p case.

Contribution

It introduces new results on the conjecture's pro-p version and higher-dimensional analogues, linking several open problems in group theory.

Findings

Verified the pro-p version of the conjecture.

Established connections between deficiency stabilization and the D2 Problem.

Extended analysis to higher-dimensional analogues.

Abstract

We investigate a conjecture about stabilisation of deficiency in finite index subgroups and relate it to the D2 Problem of C.T.C. Wall and the Relation Gap problem. We verify the pro-$p$ version of the conjecture, as well as its higher dimensional abstract analogues.