Searcharxiv⌕ Search

arXiv subjects

François Le Maître

Publications and source records attributed to François Le Maître.

29 records · Page 2Linked to original sources
arXiv2020

Dense locally finite subgroups of Automorphism Groups of Ultraextensive Spaces

We verify a conjecture of Vershik by showing that Hall's universal countable locally finite group can be embedded as a dense subgroup in the isometry group of the Urysohn space and in the automorphism group of the random graph. In fact, we show the same for all automorphism groups of known infinite ultraextensive spaces. These include, in addition, the isometry group of the rational Urysohn space, the isometry group of the ultrametric Urysohn spaces, and the automorphism group of the universal $K_n$-free graph for all $n\geq 3$. Furthermore, we show that finite group actions on finite metric spaces or finite relational structures form a Fraïssé class, where Hall's group appears as the acting group of the Fraïssé limit. We also embed continuum many non-isomorphic countable universal locally finite groups into the isometry groups of various Urysohn spaces, and show that all dense countable subgroups of these groups are mixed identity free (MIF). Finally, we give a characterization of the isomorphism type of the isometry group of the Urysohn $Δ$-metric spaces in terms of the distance value set $Δ$.

math.LO↗
arXiv2018

On a measurable analogue of small topological full groups

We initiate the study of a measurable analogue of small topological full groups that we call $\mathrm L^1$ full groups. These groups are endowed with a Polish group topology which admits a natural complete right invariant metric. We mostly focus on $\mathrm L^1$ full groups of measure-preserving $\mathbb Z$-actions which are actually a complete invariant of flip conjugacy. We prove that for ergodic actions the closure of the derived group is topologically simple although it can fail to be simple. We also show that the closure of the derived group is connected, and that for measure-preserving free actions of non-amenable groups the closure of the derived group and the $\mathrm L^1$ full group itself are never amenable. In the case of a measure-preserving ergodic $\mathbb Z$-action, the closure of the derived group is shown to be the kernel of the index map. If such an action is moreover by homeomorphism on the Cantor space, we show that the topological full group is dense in the $\mathrm L^1$ full group. Using Juschenko-Monod and Matui's results on topological full groups, we conclude that $\mathrm L^1$ full groups of ergodic $\mathbb Z$-actions are amenable as topological groups, and that they are topologically finitely generated if and only if the $\mathbb Z$-action has finite entropy.

math.DS↗
arXiv2017

The number of topological generators for full groups of ergodic equivalence relations

We completely elucidate the relationship between two invariants associated with an ergodic probability measure-preserving (pmp) equivalence relation, namely its cost and the minimal number of topological generators of its full group. It follows that for any free pmp ergodic action of the free group on $n$ generators, the minimal number of topological generators for the full group of the action is $n+1$, answering a question of Kechris.

math.DS↗
arXiv2017

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.

math.GR↗
arXiv2016

Infinitesimal generators and quasi non-archimedean topological groups

We show that connected separable locally compact groups are infinitesimally finitely generated, meaning that there is an integer $n$ such that every neighborhood of the identity contains $n$ elements generating a dense subgroup. We generalize a theorem of Schreier and Ulam by showing that any separable connected compact group is infinitesimally $2$-generated. Inspired by a result of Kechris, we introduce the notion of a quasi non-archimedean group. We observe that full groups are quasi non-archimedean, and that every continuous homomorphism from an infinitesimally finitely generated group into a quasi non-archimedean group is trivial. We prove that a locally compact group is quasi non-archimedean if and only if it is totally disconnected, and provide various examples which show that the picture is much richer for Polish groups. In particular, we get an example of a Polish group which is infinitesimally $1$-generated but totally disconnected, strengthening Stevens' negative answer to Problem 160 from the Scottish book.

math.GR↗
arXiv2016

On strongly just infinite profinite branch groups

For profinite branch groups, we first demonstrate the equivalence of the Bergman property, uncountable cofinality, Cayley boundedness, the countable index property, and the condition that every non-trivial normal subgroup is open; compact groups enjoying the last condition are called strongly just infinite. For strongly just infinite profinite branch groups with mild additional assumptions, we verify the invariant automatic continuity property and the locally compact automatic continuity property. Examples are then presented, including the profinite completion of the first Grigorchuk group. As an application, we show that many Burger-Mozes universal simple groups enjoy several automatic continuity properties.

math.GR↗
arXiv2016

Orbit full groups for locally compact groups

We show that the topological rank of an orbit full group generated by an ergodic, probability measure-preserving free action of a non-discrete unimodular locally compact Polish group is two. For this, we use the existence of a cross section and show that for a locally compact Polish group, the full group generated by any dense subgroup is dense in the orbit full group of the action of the group. We prove that the orbit full group of a free action of a locally compact Polish group is extremely amenable if and only if the acting group is amenable, using the fact that the full group generates the von Neumann algebra of the action.

math.GR↗
arXiv2015

Connected Polish groups with ample generics

In this paper, we give the first examples of connected Polish groups that have ample generics, answering a question of Kechris and Rosendal. We show that any Polish group with ample generics embeds into a connected Polish group with ample generics and that full groups of type III hyperfinite ergodic equivalence relations have ample generics. We also sketch a proof of the following result: the full group of any type III ergodic equivalence relation has topological rank 2.

math.GR↗
arXiv2015

On full groups of non ergodic probability measure preserving equivalence relations

We give a formula relating the topological rank of the full group of an aperiodic pmp equivalence relation to the cost of its ergodic components. Furthermore, we obtain examples of full groups having 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 give 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 to the non ergodic case a result of Giordano and Pestov.

math.DS↗
arXiv2014

More Polish full groups

We associate to every action of a Polish group on a standard probability space a Polish group that we call the orbit full group. For discrete groups, we recover the well-known full groups of pmp equivalence relations equipped with the uniform topology. However, there are many new examples, such as orbit full groups associated to measure preserving actions of locally compact groups. In fact, we show that such full groups are complete invariants of orbit equivalence. We give various characterizations of the existence of a dense conjugacy class for orbit full groups, and we show that the ergodic ones actually have a unique Polish group topology. Furthermore, we characterize ergodic full groups of countable pmp equivalence relations as those admitting non-trivial continuous character representations.

math.GR↗