Dense Free Subgroups Research Articles

BASIC NONARCHIMEDEAN JØRGENSEN THEORY

Abstract We prove a nonarchimedean analogue of Jørgensen’s inequality, and use it to deduce several algebraic convergence results. As an application, we show that every dense subgroup of ${\mathrm {SL}_2}(K)$ , where K is a p-adic field, contains two elements that generate a dense subgroup of ${\mathrm {SL}_2}(K)$ , which is a special case of a result by Breuillard and Gelander [‘On dense free subgroups of Lie groups’, J. Algebra261(2) (2003), 448–467]. We also list several other related results, which are well known to experts, but not easy to locate in the literature; for example, we show that a nonelementary subgroup of ${\mathrm {SL}_2}(K)$ over a nonarchimedean local field K is discrete if and only if each of its two-generator subgroups is discrete.

Strongly dense free subgroups of semisimple algebraic groups II

It was shown in [9] that there exist strongly dense free subgroups in any semisimple algebraic group over a large enough field. These are nonabelian free subgroups all of whose subgroups are either cyclic or Zariski-dense. Here we show that the same is true for as long as the transcendence degree of the field is at least 1 in characteristic 0 and at least 2 in positive characteristic. We also consider related questions for surface groups.

On profinite groups with positive rank gradient

We prove that a profinite group G with positive rank gradient does not satisfy a group law. In the case when G is a pro- p group, we show that G contains a nonabelian dense free subgroup.

Dense free subgroups of automorphism groups of homogeneous partially ordered sets

Abstract A countable poset is ultrahomogeneous if every isomorphism between its finite subposets can be extended to an automorphism. If A is such a poset, then the group Aut ⁡ ( A ) {\operatorname{Aut}(A)} has a natural topology in which Aut ⁡ ( A ) {\operatorname{Aut}(A)} is a Polish topological group. We consider the problem of whether Aut ⁡ ( A ) {\operatorname{Aut}(A)} contains a dense free subgroup of two generators. We show that if A is ultrahomogeneous, then Aut ⁡ ( A ) {\operatorname{Aut}(A)} contains such a subgroup. Moreover, we characterize those countable ultrahomogeneous posets A such that for each natural number m the set of all cyclically dense elements g ¯ ∈ Aut ( A ) m {\overline{g}\in\operatorname{Aut}(A)^{m}} for the diagonal action is comeager in Aut ( A ) m {\operatorname{Aut}(A)^{m}} . In our considerations we strongly use the result of Schmerl which says that there are essentially four types of countably infinite ultrahomogeneous posets.

Highly faithful actions and dense free subgroups in full groups

In this paper, we show that every measure-preserving ergodic equivalence relation of cost less than m comes from a “rich” faithful invariant random subgroup of the free group on m generators, strengthening a result of Bowen which had been obtained by a Baire category argument. Our proof is completely explicit: we use our previous construction of topological generators for full groups and observe that these generators induce a totally non free action. We then twist this construction so that the action is moreover amenable onto almost every orbit and highly faithful. In particular, we obtain that the full group of a measure-preserving ergodic equivalence of cost less than m contains a dense free subgroup on m generators.

On full groups of non-ergodic probability-measure-preserving equivalence relations

This article generalizes our previous results [Le Maître. The number of topological generators for full groups of ergodic equivalence relations. Invent. Math. 198 (2014), 261–268] to the non-ergodic case by giving a formula relating the topological rank of the full group of an aperiodic probability-measure-preserving (pmp) equivalence relation to the cost of its ergodic components. Furthermore, we obtain examples of full groups that have a dense free subgroup whose rank is equal to the topological rank of the full group, using a Baire category argument. We then study the automatic continuity property for full groups of aperiodic equivalence relations, and find a connected metric for which they have the automatic continuity property. This allows us to provide an algebraic characterization of aperiodicity for pmp equivalence relations, namely the non-existence of homomorphisms from their full groups into totally disconnected separable groups. A simple proof of the extreme amenability of full groups of hyperfinite pmp equivalence relations is also given, generalizing a result of Giordano and Pestov to the non-ergodic case [Giordano and Pestov. Some extremely amenable groups related to operator algebras and ergodic theory. J. Inst. Math. Jussieu6(2) (2007), 279–315, Theorem 5.7].

On dense free subgroups of the automorphism group

We obtain results on existence of dense free subgroups of the automorphism group of a homogeneous model.

Strongly dense free subgroups of semisimple algebraic groups

We show that (with one possible exception) there exist strongly dense free subgroups in any semisimple algebraic group over a large enough field. These are nonabelian free subgroups all of whose subgroups are either cyclic or Zariski dense. As a consequence, we get new generating results for finite simple groups of Lie type and a strengthening of a theorem of Borel related to the Hausdorff-Banach-Tarski paradox. In a sequel to this paper, we use this result to also establish uniform expansion properties for random Cayley graphs over finite simple groups of Lie type

Groups where free subgroups are abundant

Given an infinite topological group G and a cardinal κ > 0 , we say that G is almost κ-free if the set of κ-tuples ( g i ) i ∈ κ ∈ G κ which freely generate free subgroups of G is dense in G κ . In this note we examine groups having this property and construct examples. For instance, we show that if G is a non-discrete Hausdorff topological group that contains a dense free subgroup of rank κ > 0 , then G is almost κ-free. A consequence of this is that for any infinite set Ω, the group of all permutations of Ω is almost 2 | Ω | -free. We also show that an infinite topological group is almost ℵ 0 -free if and only if it is almost n-free for each positive integer n. This generalizes the work of Dixon, and Gartside and Knight.

Prographes sylvestres et groupes profinis presque libres

In this article we want to give an analogous in the profinite case to the following theorem: an abstract group is free if and only if it acts freely on a tree. In a first time we define a combinatory object, the protrees, which are particular inductive systems extracted from projective systems of graphs. Then we define a notion of profinite action. These objects allow us to give the following analogous: a profinite group contains a dense abstract free subgroup if and only if it acts profreely on a protree.

A topological Tits alternative

Let k be a local eld, and GLn(k) a linear group over k. We prove that either contains a relatively open solvable subgroup, or it contains a relatively dense free subgroup. This result has applications in dynamics, Riemannian foliations and pronite groups.

Dimension and randomness in groups acting on rooted trees

We explore the structure of thepp-adic automorphism groupYYof the infinite rooted regular tree. We determine the asymptotic order of a typical element, answering an old question of Turán. We initiate the study of a general dimension theory of groups acting on rooted trees. We describe the relationship between dimension and other properties of groups such as solvability, existence of dense free subgroups and the normal subgroup structure. We show that subgroups ofWWgenerated by three random elements are full dimensional and that there exist finitely generated subgroups of arbitrary dimension. Specifically, our results solve an open problem of Shalev and answer a question of Sidki.

On dense free subgroups of Lie groups

We give a method for constructing dense and free subgroups in real Lie groups. In particular we show that any dense subgroup of a connected semisimple real Lie group G contains a free group on two generators which is still dense in G , and that any finitely generated dense subgroup in a connected non-solvable Lie group H contains a dense free subgroup of rank ⩽2·dim H . This answers a question of Carriere and Ghys, and it gives an elementary proof to a conjecture of Connes and Sullivan on amenable actions, which was first proved by Zimmer.

Residual properties of groups and probabilistic methods

We outline a probabilistic approach to the solution of a number of problems concerning residual properties of infinite groups. This approach gives rise to a new and short proof of a well known conjecture of Magnus. It also yields new results on residual properties of the modular group PSL 2( Z) and of other free products of finite groups. Similar ideas can be used to show the existence of finitely generated dense free subgroups in various profinite groups, and yield an analogue of the Tits alternative for profinite completions of linear groups.

Finitely generated profinitely dense free groups in higher rank semi-simple groups

We prove that if Γ is an arithmetic subgroup of a non-compact linear semi-simple groupG such that the associated simply connected algebraic group over ℚ has the so-called congruence subgroup property, then Γ contains a finitely generated profinitely dense free subgroup. As a corollary we obtain af·g·p·d·f subgroup of SL n (ℤ) (n ≧ 3. More generally, we prove that if Γ is an irreducible arithmetic non-cocompact lattice in a higher rank group, then Γ containsf·g·p·d·f groups.

Aut (M ) Has a Large Dense Free Subgroup for Saturated M

Aut (<i>M</i> ) Has a Large Dense Free Subgroup for Saturated <i>M</i>