Co-rank and Betti number of a group

TL;DR

This paper investigates the possible combinations of co-rank and Betti number in finitely generated groups, demonstrating their realizability and implications for topology.

Contribution

It establishes that any feasible combination of co-rank and Betti number can be realized by some finitely presented group, advancing understanding in topology.

Findings

Any combination of co-rank and Betti number within constraints is realizable.

Results have implications for manifold and foliation topology.

Provides a classification framework for group invariants.

Abstract

We study the maximal ranks of a free and a free abelian quotients of a finitely generated group, called co-rank (inner rank, cut number) and the Betti number, respectively. We show that any combination of these values within obvious constraints is realized for some finitely presented group, which is important for manifold and foliation topology.