Finite groups of arbitrary deficiency

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.

7 pages, 1 figure; v2 minor changes, to appear in the Bulletin of the London Mathematical Society

References in corpus (2)

• Deficiency and abelianized deficiency of some virtually free groups
• The deficiencies of Kaehler groups