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We continue our study of the perfect kernel of the space of transitive actions of Baumslag-Solitar groups by investigating high transitivity. We show that actions of finite phenotype are never highly transitive, except when the phenotype is 1, in which case high transitivity is actually generic. In infinite phenotype, high transitivity is generic, except when |m|=|n| where it is empty. We also reinforce the dynamical properties of the action by conjugation on the perfect kernel that we had established in our first paper, replacing topological transitivity by high topological transitivity., Comment: A continuation of "On the space of subgroups of Baumslag-Solitar groups I: perfect kernel and phenotype", by Alessandro Carderi, Damien Gaboriau, Fran\c{c}ois Le Ma\^itre, Yves Stalder (arXiv:2210.14990). 44 pages, 1 figure

We introduce several approaches to studying the Cantor-Bendixson decomposition of and the dynamics on the (topological) space of subgroups for various families of countable groups. In particular, we uncover the perfect kernel and the Cantor-Bendixson rank of the space of subgroups of many new groups, including for instance infinitely ended groups, limit groups, hyperbolic 3-manifold groups and many graphs of groups. We also study the topological dynamics of the conjugation action on the perfect kernel, establishing the conditions for topological transitivity and higher topological transitivity. As an application, we obtain many new examples of groups in the class A of Glasner and Monod, i.e. admitting faithful transitive amenable actions. This includes for example right-angled Artin groups, limit groups, finitely presented C'(1/6) small cancellation groups, random groups at density d<1/6, and more generally all virtually compact special groups., Comment: Results about subgroups with Schreier graph quasi-isometric with Z extended to quasi-isometry with N. Some typos corrected and some cosmetic changes

Given a Baumslag-Solitar group, we study its space of subgroups from a topological and dynamical perspective. We first determine its perfect kernel (the largest closed subset without isolated points). We then bring to light a natural partition of the space of subgroups into one closed subset and countably many open subsets that are invariant under the action by conjugation. One of our main results is that the restriction of the action to each piece is topologically transitive. This partition is described by an arithmetically defined function, that we call the phenotype, with values in the positive integers or infinity. We eventually study the closure of each open piece and also the closure of their union. We moreover identify in each phenotype a (the) maximal compact invariant subspace., Comment: New version including the referee's suggestions. 63 pages

Let $n\ge 3$, and let $\mathrm{Out}(W_n)$ be the outer automorphism group of a free Coxeter group $W_n$ of rank $n$. We study the growth of the dimension of the homology groups (with coefficients in any field $\mathbb{K}$) along Farber sequences of finite-index subgroups of $\mathrm{Out}(W_n)$. We show that, in all degrees up to $\lfloor\frac{n}{2}\rfloor-1$, these Betti numbers grow sublinearly in the index of the subgroup. When $\mathbb{K}=\mathbb{Q}$, through L\"uck's approximation theorem, this implies that all $\ell^2$-Betti numbers of $\mathrm{Out}(W_n)$ vanish up to degree $\lfloor\frac{n}{2}\rfloor-1$. In contrast, in top dimension equal to $n-2$, an argument of Gaboriau and No\^us implies that the $\ell^2$-Betti number does not vanish. We also prove that the torsion growth of the integral homology is sublinear. Our proof of these results relies on a recent method introduced by Ab\'ert, Bergeron, Fr\k{a}czyk and Gaboriau. A key ingredient is to show that a version of the complex of partial bases of $W_n$ has the homotopy type of a bouquet of spheres of dimension $\lfloor\frac{n}{2}\rfloor-1$.

We prove new vanishing results on the growth of higher torsion homologies for suitable arithmetic lattices, Artin groups and mapping class groups. The growth is understood along Farber sequences, in particular, along residual chains. For principal congruence subgroups, we also obtain strong asymptotic bounds for the torsion growth. As a central tool, we introduce a quantitative homotopical method called effective rebuilding. This constructs small classifying spaces of finite index subgroups, at the same time controlling the complexity of the homotopy. The method easily applies to free abelian groups and then extends recursively to a wide class of residually finite groups., Comment: 51 pages, 3 figures. Modifications after referee's recommandations. A section added of "1.3-speculations and questions" relating the homology growth with the sofic entropy Betti numbers. Name change: Right-angled groups become Chain-commuting groups (see Note 1, p. 8). Comments and a new reference added for the proof of Proposition 10.15 that was incomplete. To be published in J. Eur. Math. Soc

We show that several new classes of groups are measure strongly treeable. In particular, finitely generated groups admitting planar Cayley graphs, elementarily free groups, and the group of isometries of the hyperbolic plane and all its closed subgroups. In higher dimensions, we also prove a dichotomy that the fundamental group of a closed aspherical 3-manifold is either amenable or has strong ergodic dimension 2. Our main technical tool is a method for finding measurable treeings of Borel planar graphs by constructing one-ended spanning subforests in their planar dual. Our techniques for constructing one-ended spanning subforests also give a complete classification of the locally finite pmp graphs which admit Borel a.e. one-ended spanning subforests., Comment: 42 pages, 2 x 2 figures

We study probability measure preserving (p.m.p.) non-free actions of free groups and the associated IRS's. The perfect kernel of a countable group Gamma is the largest closed subspace of the space of subgroups of Gamma without isolated points. We introduce the class of totipotent ergodic p.m.p. actions of Gamma: those for which almost every point-stabilizer has dense conjugacy class in the perfect kernel. Equivalently, the support of the associated IRS is as large as possible, namely it is equal to the whole perfect kernel. We prove that every ergodic p.m.p. equivalence relation R of cost $

The purpose of this note is to introduce a trick which relates the (non)-vanishing of the top-dimensional $\ell^2$-Betti numbers of actions with that of sub-actions. We provide three different types of applications: we prove that the $\ell^2$-Betti numbers of Aut($F_n$) and Out($F_n$) (and of their Torelli subgroups) do not vanish in degree equal to their virtual cohomological dimension, we prove that the subgroups of the 3-manifold groups have vanishing $\ell^2$-Betti numbers in degree 3 and 2 and we prove for instance that $F_2^d \times Z$ has ergodic dimension $d + 1$., Comment: ''Camille No\^us'' is a scientific consortium created to affirm the collaborative and open nature of knowledge creation and dissemination, under the control of the academic community. This scientific collective, like Bourbaki, H. P. de Saint Gervais or A. Besse in mathematics takes on the identity of a scientific personality who embodies the collective contribution of the academic community

We consider probability measure preserving discrete groupoids, group actions and equivalence relations in the context of general probability spaces. We study for these objects the notions of cost, $\beta$-invariant and some higher-dimensional variants. We also propose various convergence results about $\ell^2$-Betti numbers and rank gradient for sequences of actions, groupoids or equivalence relations under weak finiteness assumptions. In particular we connect the combinatorial cost with the cost of the ultralimit equivalence relations. Finally a relative version of Stuck-Zimmer property is also considered., Comment: 64 pages; v2 and v3 small corrections and precisions added; final version, to appear in the Annales Henri Lebesgue

The entropy in dynamical systems was introduced by A. Kolmogorov. Initially dedicated to iterations of one finite measure preserving transformation, the notion was gradually generalized so as to encompass amenable group actions and topological actions. L. Bowen (2008) succeeded in breaking the non-amenable frontier by introducing the sofic entropy. This invariant provides the same services as the classical entropy for the measured actions of the sofic groups (a class which contains the residually finite groups). In 2010, D. Kerr et H. Li established a topological version together with a variational principle., Comment: in French

Let $G$ be either a profinite or a connected compact group, and $\Gamma, \Lambda$ be finitely generated dense subgroups. Assuming that the left translation action of $\Gamma$ on $G$ is strongly ergodic, we prove that any cocycle for the left-right translation action of $\Gamma\times\Lambda$ on $G$ with values in a countable group is virtually cohomologous to a group homomorphism. Moreover, we prove that the same holds if $G$ is a (not necessarily compact) connected simple Lie group provided that $\Lambda$ contains an infinite cyclic subgroup with compact closure. We derive several applications to OE - and W$^*$- superrigidity. In particular, we obtain the first examples of compact actions of $\mathbb F_2\times\mathbb F_2$ which are W$^*$-superrigid.

In 1987, Ornstein and Weiss discovered that the Bernoulli $2$-shift over the rank two free group factors onto the seemingly larger Bernoulli $4$-shift. With the recent creation of an entropy theory for actions of sofic groups (in particular free groups), their example shows the surprising fact that entropy can increase under factor maps. In order to better understand this phenomenon, we study a natural generalization of the Ornstein--Weiss map for countable groups. We relate the increase in entropy to the cost and to the first $\ell^2$-Betti number of the group. More generally, we study coboundary maps arising from simplicial actions and, under certain assumptions, relate $\ell^2$-Betti numbers to the failure of the Juzvinski{\u\i} addition formula. This work is built upon a study of entropy theory for algebraic actions. We prove that for actions on profinite groups via continuous group automorphisms, topological sofic entropy is equal to measure sofic entropy with respect to Haar measure whenever the homoclinic subgroup is dense. For algebraic actions of residually finite groups we find sufficient conditions for the sofic entropy to be equal to the supremum exponential growth rate of periodic points.

The goal of this note is to announce certain results in orbit equivalence theory, especially concerning the approximation of p.m.p. standard equivalence relations by increasing sequence of sub-relations, with applications to the behavior of the Bernoulli percolation on graphs at the threshold pu. R\'esum\'e en Fran\c{c}ais. Approximations de relations d'\'equivalence standards et percolation de Bernoulli \`a p\_u.

Two-page paper on the notion of cost of groups and measured equivalence relations to appear in the "What is?" series in the Notices of the AMS., Comment: 2 pages

We give a survey of various recent developments in orbit equivalence and measured group theory. This subject aims at studying infinite countable groups through their measure preserving actions., Comment: 2010 Hyderabad ICM proceeding; Dans Proceedings of the International Congress of Mathematicians, Hyderabad, India - International Congress of Mathematicians (ICM), Hyderabad : India (2010)

We study the analogue in orbit equivalence of free product decomposition and free indecomposability for countable groups. We introduce the (orbit equivalence invariant) notion of freely indecomposable ({\FI}) standard probability measure preserving equivalence relations and establish a criterion to check it, namely non-hyperfiniteness and vanishing of the first $L^2$-Betti number. We obtain Bass-Serre rigidity results, \textit{i.e.} forms of uniqueness in free product decompositions of equivalence relations with ({\FI}) components. The main features of our work are weak algebraic assumptions and no ergodicity hypothesis for the components. We deduce, for instance, that a measure equivalence between two free products of non-amenable groups with vanishing first $\ell^2$-Betti numbers is induced by measure equivalences of the components. We also deduce new classification results in Orbit Equivalence and II$_1$ factors., Comment: minor additions

We show that every non-amenable free product of groups admits free ergodic probability measure preserving actions which have relative property (T) in the sense of S.-Popa \cite[Def. 4.1]{Pop06}. There are uncountably many such actions up to orbit equivalence and von Neumann equivalence, and they may be chosen to be conjugate to any prescribed action when restricted to the free factors. We exhibit also, for every non-amenable free product of groups, free ergodic probability measure preserving actions whose associated equivalence relation has trivial outer automorphisms group. This gives in particular the first examples of such actions for the free group on $2$ generators., Comment: Improvement in the presentation of the replacement trick. Introduction of compressions for non-ergodic equivalence relations

We give a positive answer, in the measurable-group-theory context, to von Neumann's problem of knowing whether a non-amenable countable discrete group contains a non-cyclic free subgroup. We also get an embedding result of the free-group von Neumann factor into restricted wreath product factors., Comment: 10 pages, to appear in Inventiones Mathematicae

Measure Equivalence (ME) is the measure theoretic counterpart of quasi-isometry. This field grew considerably during the last years, developing tools to distinguish between different ME classes of countable groups. On the other hand, contructions of ME equivalent groups are very rare. We present a new method, based on a notion of measurable free-factor, and we apply it to exhibit a new family of groups that are measure equivalent to the free group. We also present a quite extensive survey on results about Measure Equivalence for countable groups., Comment: To appear in Ergodic Theory and Dynamical Systems

The main goal of this paper is to answer question 1.10 and settle conjecture 1.11 of Benjamini-Lyons-Schramm [BLS99] relating harmonic Dirichlet functions on a graph to those of the infinite clusters in the uniqueness phase of Bernoulli percolation. We extend the result to more general invariant percolations, including the Random-Cluster model. We prove the existence of the nonuniqueness phase for the Bernoulli percolation (and make some progress for Random-Cluster model) on unimodular transitive locally finite graphs admitting nonconstant harmonic Dirichlet functions. This is done by using the device of $\ell^2$ Betti numbers., Comment: to appear in Geometric And Functional Analysis (GAFA)

For each $2 \leq n \leq \infty$, we construct an uncountable family of free ergodic measure preserving actions $\alpha_t$ of the free group $\Bbb F_n$ on the standard probability space $(X, \mu)$ such that any two are non orbit equivalent (in fact, not even stably orbit equivalent). These actions are all ``rigid'' (in the sense of [Po01]), with the II$_1$ factors $L^\infty(X, \mu)\rtimes_{\alpha_t} \Bbb F_n$ mutually non stably isomorphic (even non-stably isomorphic) and in the class $\Cal H\Cal T_{_{s}}.$, Comment: New version (13 pages), which includes the case $ n = \infty$; revised version 04/08/04