L^2-Betti numbers of one-relator groups

Abstract

We determine the L^2-Betti numbers of all one-relator groups and all surface-plus-one-relation groups (surface-plus-one-relation groups were introduced by Hempel who called them one-relator surface groups). In particular we show that for all such groups G, the L^2-Betti numbers b_n^{(2)}(G) are 0 for all n>1. We also obtain some information about the L^2-cohomology of left-orderable groups, and deduce the non-L^2 result that, in any left-orderable group of homological dimension one, all two-generator subgroups are free.

18 pages, version 3, minor changes. To appear in Math. Ann

Cited by in corpus (6)

• The strong Atiyah and Lück approximation conjectures for one-relator groups
• Universal $L^2$-torsion, polytopes and applications to $3$-manifolds
• Non-orientable surface-plus-one-relation groups
• The agrarian polytope of two-generator one-relator groups
• Word images in symmetric and classical groups of Lie type are dense
• Vanishing of $L^2$-Betti numbers and failure of acylindrical hyperbolicity of matrix groups over rings