# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
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:
 20F05 Generators, relations, and presentations of groups 20E26 Residual properties of groups and generalizations; residually finite groups 43A07 Means on groups, semigroups, etc.; amenable groups 20E18 Limits, profinite groups 20F65 Geometric group theory
Full Text:
##### References:
 [1] Abels, H.: An example of a finitely presentable solvable group. In: ”Homological Group Theory, Proceedings Durham 1977”. London Math. Soc. Lecture Note Ser. vol. 36, pp. 105--211 (1979) [2] Abels, H.: ”Finite presentability of S-arithmetic groups. Compact presentability of solvable groups”. Lecture Notes in Math., vol. 1261. Springer, Berlin (1987) · Zbl 0621.20015 [3] Burger M., Mozes S.: Lattices in product of trees. Publ. Math. Inst. Hautes Études Sci. 92, 151--194 (2000) · Zbl 1007.22013 · doi:10.1007/BF02698916 [4] Cornulier Y.: Finitely presentable, non-Hopfian groups with Kazhdan’s Property and infinite outer automorphism group. Proc. Am. Math. Soc. 135, 951--959 (2007) · Zbl 1184.20035 · doi:10.1090/S0002-9939-06-08588-1 [5] Cornulier Y., Guyot L., Pitsch W.: On the isolated points in the space of groups. J. Algebra 307(1), 254--277 (2007) · Zbl 1132.20018 · doi:10.1016/j.jalgebra.2006.02.012 [6] Chabauty C.: Limite d’ensembles et géométrie des nombres. Bull. Soc. Math. France 78, 143--151 (1950) · Zbl 0039.04101 [7] Champetier C.: L’espace des groupes de type fini. Topology 39(4), 657--680 (2000) · Zbl 0959.20041 · doi:10.1016/S0040-9383(98)00063-9 [8] Champetier C., Guirardel V.: Limit groups as limits of free groups. Israel J. Math. 146, 1--75 (2005) · Zbl 1103.20026 · doi:10.1007/BF02773526 [9] Elek G., Szabo E.: On sofic groups. J. Group Theory 9(2), 161--171 (2006) · Zbl 1153.20040 · doi:10.1515/JGT.2006.011 [10] Grigorchuk R.: Degrees of growth of finitely generated groups and the theory of invariant means. Izv. Akad. Nauk SSSR Ser. Mat. 48(5), 939--985 (1984) [11] Gromov M.: Endomorphisms of symbolic algebraic varieties. J. Eur. Math. Soc. 1(2), 109--197 (1999) · Zbl 0998.14001 · doi:10.1007/PL00011162 [12] Pestov V.: Hyperlinear and sofic groups: a brief guide. Bull. Symb. Logic 14(4), 449--480 (2008) · Zbl 1206.20048 · doi:10.2178/bsl/1231081461 [13] Thom A.: Examples of hyperlinear groups without factorization property. Groups Geom. Dyn. 4(1), 195--208 (2010) · Zbl 1187.22002 · doi:10.4171/GGD/80 [14] Weiss B.: Sofic groups and dynamical systems. Ergodic theory and harmonic analysis (Mumbai, 1999). Sankhyā Ser. A 62(3), 350--359 (2000)