Rank 1 phenomena for 1-relator and deficiency 1 groups

TL;DR

The paper investigates two rank-1 phenomena in finitely presented groups: (i) infinite dimensional $H^2_b(G;\mathbb{R})$ via homogeneous quasimorphisms, and (ii) the existence of a finite index subgroup whose pro-$p$ completion is not $p$-adic analytic. It proves that every non-small group with deficiency at least one, in particular many 1-relator groups, satisfies at least one of these phenomena, and uses this to derive strong conclusions about bounded generation. A central result is that among 1-relator groups, the boundedly generated ones are exactly the cyclic groups and the solvable Baumslag–Solitar groups $BS(1,n)$; otherwise the rank-1 dichotomy applies, via either infinite bounded cohomology or non-$p$-adic analytic pro-$p$ completions. The methods combine bounded cohomology, pro-$p$ techniques, and the structure of HNN extensions and free-by-cyclic groups to map out a nearly complete picture, with precise exclusions and several conjectural gaps surrounding strictly ascending HNN extensions. This work offers a framework for understanding how rank-1 phenomena govern the large-scale algebraic and subgroup-growth behavior of deficiency-one groups, especially 1-relator groups, with implications for acylindrical hyperbolicity and linearity questions.

Abstract

We examine second bounded cohomology and mod p homology in finite index subgroups of 1-relator groups and groups with a presentation of deficiency at least one. We use this to determine exactly which 1-relator groups are boundedly generated, as well as the groups of deficiency at least one up to a class of groups that conjecturally do not exist.

Theorems & Definitions (31)

• Theorem 2.1
• Corollary 2.2
• proof
• Theorem 2.3
• Corollary 2.4
• Corollary 2.5
• Theorem 2.6
• proof
• Theorem 3.1
• proof