Weak forms of amenability for one-relator groups
The authors discuss weak amenability on discrete groups \(G\). The authors call a group `\(a\)-\(T\)-moyennable' if it satisfies the condition that when there exists a sequence of normalised positive definite functions on \(G\), tending to 0 at infinity, the sequence converges pointwise to the constant function 1. This condition is not really an amenability condition, but mimics a property of all separable locally compact groups which follows from a result of P. Urysohn (circa 1924). The term \(a\)-\(T\)-moyennable was used by \textit{M. Gromov} [Asymptotic invariants of infinite groups, in: Geometric group theory, vol. 2 (1991; Zbl 0841.20039)] in the sense of essentially not having the property \(T\) of \textit{D. A. Kazhdan} [Funct. Anal. Appl. 1, 63-65 (1967; Zbl 0168.27602)]. \(G\) is called inner amenable if there is a finitely additive mean with total mass 1 on \(G -\{e\}\) which is invariant under inner automorphisms. \(G\) is usually assumed as having two generators and one relator, say \(G=\langle a,b:R\rangle\). Let \(N\) denote the normal subgroup generated by one of the generators which is assumed to appear with exponent sum 0. As in M. Dehn's Freiheitssatz [\textit{W. Magnus}, J. Reine Angew. Math. 163, 141-165 (1930; JFM 56.0134.03)], the graph of \(G\) is divided into `layers' of amalgams of freely generated subgroups and \(N\) expressed as the union of these. The authors use only the simplest situation [see Ch.~II.5 of \textit{B. Chandler} and \textit{W. Magnus}, The history of combinatorial group theory (Berlin etc. 1982; Zbl 0498.20001)] showing that \(N\) is freely generated whenever \(G\) has nontrivial centre. Then, by results quoted from \textit{P. Jolissaint, P. Julg} and \textit{A. Valette} in ``Locally compact groups with the Haagerup property'' (to appear), \(G/N\) is amenable and so also \(G\) is \(a\)-\(T\)-moyennable. Taking \(G\) to be an HNN extension [see \textit{G. Higman, B. H. Neumann} and \textit{H. Neumann}, J. Lond. Math. Soc. 24, 247-254 (1950; Zbl 0034.30101)]\( (H,A,B,\theta)\) where \(\theta\) is a homomorphism \(s \mapsto sas^{-1}\) from \(A\) to \(B\), (usually both) proper subgroups of \(H\). The authors assume that \(G\) acts, without inversion and with orientation given by \(gH\), by left translation on the tree \(X\) whose vertices constitute \(G/H\) and arrows constitute \( G/A\); \(H/A\) indexes the arrows entering vertices, \(H/B\) the arrows leaving vertices. The boundary \(\partial X\) consists of equivalence classes of semi-infinite geodesics on the tree. The authors require \(G\) to act faithfully as transformation-group on the disjoint union \(X \sqcup \partial X\) and (presumably) install a metric structure on \(X\) so that \(G\) acts as homeomorphisms on the Hausdorff topological space \(X \sqcup \partial X\) denoted by \(\Omega\) [cf. \textit{E. Ghys} and \textit{P. de la Harpe}, Sur les groupes hyperboliques d'après Mikhael Gromov, Prog. Math. 83 (Boston 1990; Zbl 0731.20025)]. It is known for the action of \(G\) on a tree by an automorphism that it is either elliptic (there exist fixed points) or hyperbolic (exactly two fixed points on the boundary, a source and a sink) [cf. \textit{J.-P. Serre}, Arbres, amalgames, \(\text{SL}_2\), Asterisque 46 (1977; Zbl 0369.20013)]. The authors wish to apply a result of \textit{E. Bédos} and \textit{P. de la Harpe} [Enseign. Math. (2) 32, 139-157 (1986; Zbl 0605.43002)] to prove that \(G\) is not inner amenable. They need to ensure that G acts faithfully on \(\Omega\), has transverse hyperbolic elements and that there is an equivariant map \( \delta:{G - \{ {e} \}} \to \Omega\). To ensure that there are transverse hyperbolic elements it is supposed that \(A \neq H \neq B\). Faithfulness is here equivalent to elements of \(G\) fixing all the vertices. The paper is faulty regarding the definition of \({\widetilde{\gamma}}\) on p. 142 in that the \(g\) in \(g \gamma g^{-1}\) is not specified and neither is the corresponding mapping specified in the hyperbolic case. An equivariant mapping \(\delta\) in the context of the article is such that \(\delta (g \gamma g^{-1}) = g\delta(\gamma)\). It is sufficient that the equivariant map be defined on \(X\) (and not necessarily on \(\Omega\) as presumed by the authors). Such equivariant mappings do exist. If \(\gamma\) is elliptic (i.e. \( g \gamma g^{-1} \in H\)) one sets \(\delta(\gamma)\) to be some fixed point \(gH\). For a hyperbolic \(\gamma\) the corresponding fixed point is the sink in \(\partial X\) corresponding to \(g^{n} \gamma g^{-n}\) for large enough \(n\). The authors do not verify that Proposition 7 of \textit{E. Bédos} and \textit{P. de la Harpe} (loc. cit.) is valid in their case. However this is not neccesary as non-inner-amenability follows from Bédos and de la Harpe's Proposition 1 (loc. cit.). Following an application by Bédos and de la Harpe (loc. cit.) of their Proposition 7, the authors deal with strong faithfulness on \(\partial X\), viz., for all finite sets \(F \subset G-\{e\}\) there exists an \(\omega \in \Omega\) such that \(f\omega \neq \omega\) for all \(f\) in \( F\). To be able to use this the authors ensure that the points fixed by \(g\) in \(\partial X\) are not open. In the hyperbolic case it is sufficient that \(X\) has no dead-ends and that the lengths of finite geodesic chains, with all interior vertices of degree 2, are uniformly bounded. (For the locally finite case the authors refer to the article of \textit{T. Ceccherini-Silberstein, R. Grigorchuk} and \textit{P. de la Harpe} [in Proc. Steklov Math. Inst. 224 (1999)]. In the elliptic case the sufficient condition given involves balls \(B(n)\) of radius \(n\) around some chosen vertex such that the `largest' \(n\), such that all the vertices in \(B(n)\) are in the set of fixed points in X under \(g \in G-\{e\}\), is finite. A result of \textit{P. de la Harpe} [Reduced \(C^\ast\)-algebras of discrete groups which are simple with a unique trace. Lect. Notes Math. 1132, 230-253 (1985; Zbl 0575.46049)] shows that such groups are not inner amenable and their reduced \(C^\ast\)-algebras are simple with unique trace. The authors apply this to groups with one relator and more than three generators and note that this was already proved by \textit{E. Bédos} [Operator algebras associated with HNN-extensions, 1984, unpublished] using HNN extensions. Some other of the propositions proved are also previously known. The reviewer found most of the arguments hard to decifer, several proofs less than minimal, and that complicated results are quoted without explanation or reference to origin. The reviewer presumes that the extensive use of \(\bigcap_{\gamma \in G}gAg^{-1}\) is due to a misprint.
scientific article; zbMATH DE number 1423102