The geometry of one-relator groups satisfying a polynomial isoperimetric inequality

Abstract

For every pair of positive integers p > q we construct a one-relator group R p,q whose Dehn function is n 2α where α = log 2 (2p/q). The group R p,q has no subgroup isomorphic to a Baumslag-Solitar group BS(m, n) with m = ±n, but is not automatic, not CAT(0), and cannot act freely on a CAT(0) cube complex. This answers a long-standing question on the automaticity of one-relator groups and gives counterexamples to a conjecture of Wise.

Residually finite tubular groups

“…groups with isoperimetric functions n d where d ∈ D is a dense subset of [2, ∞]. Gardam and Woodhouse showed that certain Snowflake groups embed as finite index subgroups of onerelator groups [9], and Button observed that many of these groups are not residually finite [6]. Cashen gave a quasi-isometric classification of tubular groups [7].…”

Cubulating one-relator products with torsion

“…But there are many one-relator groups for which it does not hold. The Baumslag-Solitar group xa; b | b ¡1 ab a 2 y, for example, (like many torsion free one-relator groups, see [GW19]) has Dehn function which is not linear or quadratic and thus cannot be cubulable. Thus the npXq 1 case is very interesting, but would require a more subtle statement and probably very dierent indicable, or at least torsion free.…”

Tubular groups, 1-relator groups and nonpositive curvature

“…Moreover on taking p > q ≥ 1, these are exactly the snowflake groups in [2] which have unusual Dehn functions that are greater than quadratic and so they cannot be CAT(0). Thus [13] But what about the 1-relator groups R p,q where q ≥ p ≥ 1? In this case we still have the index 2 tubular subgroup G p,q with presentation as above.…”

“…In this case we still have the index 2 tubular subgroup G p,q with presentation as above. For p = q = 1 [13] points out that G 1,1 is none other than Gersten's group. Thus R 1,1 is a 1-relator group that is virtually free by cyclic (indeed it is free by cyclic by K. S. Brown's criterion) with quadratic Dehn function and which acts freely on a CAT(0) complex but which is not CAT(0).…”

“…This was open until the construction in [13] of Gardam and Woodhouse which obtained a very close relationship between certain 2-generator 1-relator groups and certain tubular groups. More precisely, they consider the particular class of tubular groups…”