Limit Groups of Equationally Noetherian Groups

Houcine, Abderezak Ould

doi:10.1007/978-3-7643-8412-8_8

Cited by 14 publications

(10 citation statements)

References 16 publications

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“…Conversely if G is approximable by C, then G is locally fully residually-C. The same property holds also in hyperbolic groups [47,40] and more generally in equationally noetherian groups [36]. (3) Let V be a possibly infinite-dimensional vector space over a field K. Denote by GL(V, K) the group of automorphisms of V .…”

Section: Examplesmentioning

confidence: 92%

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Alternatives for pseudofinite groups

Houcine

Point

2013

Journal of Group Theory

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Abstract. The famous Tits alternative states that a linear group either contains a nonabelian free group or is soluble-by-(locally finite). In this paper we study similar alternatives in pseudofinite groups. We show, for instance, that an @ 0 -saturated pseudofinite group either contains a subsemigroup of rank 2 or is nilpotent-by-(uniformly locally finite). We call a class of finite groups G weakly of bounded rank if the radical rad.G/ has a bounded Prüfer rank and the index of the socle of G=rad.G/ is bounded. We show that an @ 0 -saturated pseudo-(finite weakly of bounded rank) group either contains a nonabelian free group or is nilpotent-by-abelian-by-(uniformly locally finite). We also obtain some relations between these kind of alternatives and amenability.

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Section: Examplesmentioning

confidence: 92%

“…Conversely if G is approximable by C, then G is locally fully residually-C. The same property holds also in hyperbolic groups ( [48,41]) and more generally in equationally noetherian groups ( [36]). …”

mentioning

confidence: 82%

Alternatives for pseudofinite groups

Houcine

Point

2013

Journal of Group Theory

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“…This coarser relation between G and H is an equivalent form of a definition due to Sela [56] that H is a G-limit group; see also [49]. A group A is a G-limit group if and only there exists a group containing A and preformed by G; see the remark after Lemma 2.18.…”

Section: Introductionmentioning

confidence: 87%

“…Baumslag, Myasnikov and Remeslinnikov show in [13,Thm B1] that all linear groups are equationally noetherian. Ould Houcine proves in [49] that, if G is equationally noetherian, then there are at most countably many groups that are limits of G, in particular, there are at most countably many groups preformed by G.…”

Section: Groups Larger or Smaller Than A Given Groupmentioning

confidence: 99%

Ordering the space of finitely generated groups

Bartholdi¹,

Erschler²

2015

Ann. inst. Fourier

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We consider the oriented graph whose vertices are isomorphism classes of finitely generated groups, with an edge from G to H if, for some generating set T in H and some sequence of generating sets S_i in G, the marked balls of radius i in (G,S_i) and in (H,T) coincide. Given a nilpotent group G, we characterize its connected component in this graph: if that connected component contains at least one torsion-free group, then it consists of those groups which generate the same variety of groups as G. The arrows in the graph define a preorder on the set of isomorphism classes of finitely generated groups. We show that a partial order can be imbedded in this preorder if and only if it is realizable by subsets of a countable set under inclusion. We show that every countable group imbeds in a group of non-uniform exponential growth. In particular, there exist groups of non-uniform exponential growth that are not residually of subexponential growth and do not admit a uniform imbedding into Hilbert space

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“…In this context, Γ-limit groups are precisely finitely generated fully residually-Γ groups, when the group Γ is either hyperbolic, as was shown by Sela in [Sel09] for the torsion-free case and by Reinfeldt and Weidmann in [RW10] for all hyperbolic groups, or Γ ∈ R, as was proved by Groves in [Gro05]. Yet more generally, the same result holds if Γ is an equationally Noetherian group, as was proved by Ould Houcine in [Hou07].…”

Section: Residual Propertymentioning

confidence: 61%

On Properties of Relatively Hyperbolic Groups

Zhang¹

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We discuss a number of problems in relatively hyperbolic groups. We show that the word problem and the conjugacy (search) problem are solvable in linear and quadratic time, respectively, for a relatively hyperbolic group, whenever the corresponding problem is solvable in linear and quadratic time in each parabolic subgroup. We also consider the class R of finitely generated toral relatively hyperbolic groups. We show that groups from R are commutative transitive and generalize a theorem proved by Baumslag in [Bau67] to R. Moreover, we discuss two definitions of (fully) residually-C groups, the classical Definition 1.2.1 and a modified Definition 1.2.5. Building upon results obtained by Ol'shanskii [Ol'93] and Osin [Osi10], we prove the equivalence of the two definitions for C = R. This is a generalization of the similar result obtained by Ol'shanskii for the class of torsion-free hyperbolic groups. Let Γ ∈ R be non-abelian and non-elementary. Kharlampovich and Miasnikov proved in [KM12] that a finitely generated fully residually-Γ group G embeds into an iterated extension of centralizers of Γ. We deduce from their theorem that every finitely generated fully residually-Γ group embeds into a group from R. On the other hand, we give an example of a finitely generated torsion-free fully residually-H group that does not embed into a group from R; H is the class of hyperbolic groups.

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Limit Groups of Equationally Noetherian Groups

Cited by 14 publications

References 16 publications

Alternatives for pseudofinite groups

Alternatives for pseudofinite groups

Ordering the space of finitely generated groups

On Properties of Relatively Hyperbolic Groups

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