On $p$-deficiency in groups

TL;DR

This paper extends the concept of $p$-deficiency in groups, proving super multiplicity for all finite index normal subgroups and exploring its implications for various classes of groups and related invariants.

Contribution

It generalizes super multiplicity of $p$-deficiency to all finite index normal subgroups and investigates its effects on group invariants and specific group classes.

Findings

Super multiplicity of $p$-deficiency holds for all finite index normal subgroups.

Some groups with non-positive $p$-deficiency have virtually positive $p$-deficiency.

Computed $p$-deficiency for Fuchsian groups and studied related invariants.

Abstract

Recently, Schlage-Puchta proved super multiplicity of $p$-deficiency for normal subgroups of $p$-power index. We extend this result to all normal subgroups of finite index. We then use the methods of the proof to show that some groups with non-positive $p$-deficiency have virtually positive $p$-deficiency. We also compute the $p$-deficiency in some cases such as Fuchsian groups and study related invariants: the lower and upper absolute $p$-homology gradients and the $p$-Euler characteristic.