A note on the growth of Betti numbers and ranks of 3-manifold groups

TL;DR

This paper explores the growth patterns of Betti numbers and ranks in the fundamental groups of certain 3-manifolds, demonstrating both fast and slow growth behaviors using recent advances in 3-manifold topology.

Contribution

It introduces cofinal filtrations of 3-manifold groups exhibiting both rapid and slow growth of Betti numbers and ranks, leveraging recent breakthroughs in the field.

Findings

Existence of cofinal filtrations with fast Betti number growth

Existence of cofinal filtrations with slow rank growth

Application of recent topological results to group growth analysis

Abstract

Let N be an irreducible, compact 3-manifold with empty or toroidal boundary which is not a closed graph manifold. Using recent work of Agol, Kahn-Markovic and Przytycki-Wise we will show that pi_1(N) admits a cofinal filtration with `fast' growth of Betti numbers as well as a cofinal filtration of pi_1(N) with `slow' growth of ranks.