## Some non-amenable groups.(English)Zbl 1264.20032

From the introduction: In this note we give the following generalisation of a result of R. M. Thomas [Math. Proc. Camb. Philos. Soc. 103, No. 3, 385-387 (1988; Zbl 0661.20017)].
Theorem 1. Let $$G$$ be a finitely generated group given by the presentation $$\langle x_1,\dots,x_d:u^{m_1}_1,\dots,u^{m_r}_r\rangle$$ such that each relator $$u_i$$ has order $$m_i$$ in $$G$$. (1) If $$G$$ is finite then $$1-d+\sum^r_{i=1}\tfrac 1{m_i}>0$$ and $$|G|\geq \tfrac 1{1-d+\sum^r_{i=1}\tfrac 1{m_i}}$$. (2) If the first $$\ell^2$$ Betti number $$\beta^2_1(G)$$ of $$G$$ is zero, then $$1-d+\sum^r_{i=1}\tfrac 1{m_i}\geq 0$$.
In particular, the case when all the exponents $$m_i$$ in the presentation are equal to 1 yields the well known observation that when the first $$\ell^2$$ Betti number is zero the deficiency of the presentation $$d-r$$ must be at most 1. The vanishing of the first $$\ell^2$$ Betti number of a group $$G$$ holds for example if $$G$$ is finite, if it satisfies Kazhdan’s property (T) or if it admits an infinite normal amenable subgroup (in particular if it is infinite amenable).

### MSC:

 20F05 Generators, relations, and presentations of groups 20J06 Cohomology of groups 43A07 Means on groups, semigroups, etc.; amenable groups

Zbl 0661.20017
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