## Euler characteristics of discrete groups, pp. 106–254 in: Groups: topological, combinatorial and arithmetic aspects

- M Chiswell
- Euler characteristics of discrete groups, pp. 106…
- 2004

- 2005

We determine the L 2-Betti numbers of all one-relator groups and all surface plus one relation groups. 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. 1 Notation and background Let G be a (discrete) group, fixed throughout the article. We use R ∪ {−∞, ∞} with the usual conventions; for example, 1 ∞ = 0, and 3 − ∞ = −∞. Let N denote the set of finite cardinals, {0, 1, 2,. . .}. We call N ∪ {∞} the set of vague cardinals, and, for each set X, we define its vague cardinal |X| ∈ N ∪ {∞} to be the cardinal of X if X is finite, and to be ∞ if X is infinite. Mappings of right modules will be written on the left of their arguments, and mappings of left modules will be written on the right of their arguments. Let C[[G]] denote the set of all functions from G to C expressed as formal sums, that is, a function a : G → C, g → a(g), will be written as g∈G a(g)g. Then C[[G]] has a natural CG-bimodule structure, and contains a copy of CG as CG-sub-bimodule. For each a ∈ C[[G]], we define a := (g∈G |a(g)| 2) 1/2 ∈ [0, ∞], and tr(a) := a(1) ∈ C. Define l 2 (G) := {a ∈ C[[G]] : a < ∞}.