# Finite groups of arbitrary deficiency

## TL;DR

This paper demonstrates that every non-positive integer can be realized as the deficiency of a finite group, specifically finite p-groups, completing the classification of deficiencies of fundamental groups of compact Kähler manifolds.

## Contribution

It proves that all non-positive integers are achievable as deficiencies of finite groups, including finite p-groups, filling a gap in the classification of Kähler manifold fundamental groups.

## Key findings

- Every non-positive integer is the deficiency of some finite p-group.
- Completes the classification of deficiencies for fundamental groups of compact Kähler manifolds.
- Shows that deficiency can be arbitrarily negative for finite groups.

## Abstract

The deficiency of a group is the maximum over all presentations for that group of the number of generators minus the number of relators. Every finite group has non-positive deficiency. We show that every non-positive integer is the deficiency of a finite group -- in fact, of a finite $p$-group for every prime $p$. This completes Kotschick's classification of the integers which are deficiencies of fundamental groups of compact Kaehler manifolds.