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).


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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