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A sofic group away from amenable groups. (English) Zbl 1247.20039
From the introduction: We give an example of a sofic group, which is not a limit of amenable groups. Let $G$ be a finitely generated group. A sequence $(G_n)$ of finitely generated groups converges to $G$ if there exists a finitely generated free group $F$, and normal subgroups $N$, $(N_n)$ of $G$ such that -- $F/N\simeq G$; $F/N_n\simeq G_n$ for all $n$; -- $(N_n)$ converges to $N$, i.e. for all $x\in N$, resp. $y\in F-N$; eventually $x\in N_n$, resp. $y\not\in N_n$. The finitely generated group $G\simeq F/N$ is isolated if whenever such a situation occurs, eventually $N_n=N$. Sofic groups were introduced by {\it M. Gromov} as “groups whose Cayley graph is initially subamenable” [J. Eur. Math. Soc. (JEMS) 1, No. 2, 109-197 (1999; Zbl 0998.14001), p. 157] and by {\it B. Weiss} [in Sankhyā, Ser. A 62, No. 3, 350-359 (2000; Zbl 1148.37302)]. It will be enough for us to know that the class of sofic groups is not empty and satisfies the following properties -- (subgroups) Subgroups of sofic groups are sofic; -- (direct limits) A group is sofic if (and only if) all its finitely generated subgroups are sofic; -- (amenable extensions) if a group is sofic-by-amenable, i.e. has an amenable quotient with sofic kernel, then it is sofic as well (in particular, amenable implies sofic); -- (marked limits) if a finitely generated group $G$ is a limit of a sequence of sofic groups $(G_n)$, then $G$ is sofic as well. In particular, residually finite groups are sofic. No group is known to fail to be sofic. The following question was asked by Gromov [loc. cit.]. Question 1. Is every finitely generated sofic group a limit of amenable groups (“initially subamenable”)? Examples of sofic groups that are not residually amenable were obtained by {\it G. Elek} and {\it E. Szabó} [in J. Group Theory 9, No. 2, 161-171 (2006; Zbl 1153.20040)], but by construction, these groups are limits of finite groups. Gromov expected a negative answer to the general question and we confirm this expectation. Theorem 2. There exists a finitely presented, non-amenable, isolated, (locally residually finite)-by-Abelian group. Corollary 3. There exists a finitely presented sofic group that is not a limit of amenable groups.

MSC:
20F05Generators, relations, and presentations of groups
20E26Residual properties of groups and generalizations; residually finite groups
43A07Means on groups, semigroups, etc.; amenable groups
20E18Limits, profinite groups
20F65Geometric group theory
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References:
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