arXiv subjects

Adrien Le Boudec

Publications and source records attributed to Adrien Le Boudec.

27 records · Page 2Linked to original sources

arXiv2017

Densely related groups

We study the class of densely related groups. These are finitely generated (or more generally, compactly generated locally compact) groups satisfying a strong negation of being finitely presented, in the sense that new relations appear at all scales. Here, new relations means relations that do not follow from relations of smaller size. Being densely related is a quasi-isometry invariant among finitely generated groups. We check that a densely related group has none of its asymptotic cones simply connected. In particular a lacunary hyperbolic group cannot be densely related. We prove that the Grigorchuk group is densely related. We also show that a finitely generated group that is (infinite locally finite)-by-cyclic and which satisfies a law must be densely related. Given a class $\mathcal{C}$ of finitely generated groups, we consider the following dichotomy: every group in $\mathcal{C}$ is either finitely presented or densely related. We show that this holds within the class of nilpotent-by-cyclic groups and the class of metabelian groups. In contrast, this dichotomy is no longer true for the class of $3$-step solvable groups.

math.GR↗

arXiv2017

Commensurated subgroups in tree almost automorphism groups

We prove that the tree almost automorphism groups admit exactly three commensurability classes of closed commensurated subgroups. Our proof utilizes an independently interesting characterization of subgroups of the tree almost automorphism groups which contain only periodic elements in terms of the dynamics of the action on the boundary of the tree. Our results further cover several interesting finitely generated subgroups of the tree almost automorphism groups, including the Thompson groups $F$, $T$, and $V$. We show in particular that Thompson's group $T$ has no commensurated subgroups other than the finite subgroups and the entire group. As a consequence, we derive several rigidity results for the possible embeddings of these groups into locally compact groups.

math.GR↗

arXiv2016

Subgroup dynamics and $C^\ast$-simplicity of groups of homeomorphisms

We study the uniformly recurrent subgroups of groups acting by homeomorphisms on a topological space. We prove a general result relating uniformly recurrent subgroups to rigid stabilizers of the action, and deduce a $C^*$-simplicity criterion based on the non-amenability of rigid stabilizers. As an application, we show that Thompson's group $V$ is $C^\ast$-simple, as well as groups of piecewise projective homeomorphisms of the real line. This provides examples of finitely presented $C^\ast$-simple groups without free subgroups. We prove that a branch group is either amenable or $C^\ast$-simple. We also prove the converse of a result of Haagerup and Olesen: if Thompson's group $F$ is non-amenable, then Thompson's group $T$ must be $C^\ast$-simple. Our results further provide sufficient conditions on a group of homeomorphisms under which uniformly recurrent subgroups can be completely classified. This applies to Thompson's groups $F$, $T$ and $V$, for which we also deduce rigidity results for their minimal actions on compact spaces.

math.GR↗

arXiv2016

C*-simplicity and the amenable radical

A countable group is C*-simple if its reduced C*-algebra is simple. It is well known that C*-simplicity implies that the amenable radical of the group must be trivial. We show that the converse does not hold by constructing explicit counter-examples. We additionally prove that every countable group embeds into a countable group with trivial amenable radical and that is not C*-simple.

math.GR↗

arXiv2016

Compact presentability of tree almost automorphism groups

We establish compact presentability, i.e. the locally compact version of finite presentability, for an infinite family of tree almost automorphism groups. Examples covered by our results include Neretin's group of spheromorphisms, as well as the topologically simple group containing the profinite completion of the Grigorchuk group constructed by Barnea, Ershov and Weigel. We additionally obtain an upper bound on the Dehn function of these groups in terms of the Dehn function of an embedded Higman-Thompson group. This, combined with a result of Guba, implies that the Dehn function of the Neretin group of the regular trivalent tree is polynomially bounded.

math.GR↗

arXiv2016

Locally compact lacunary hyperbolic groups

We investigate the class of locally compact lacunary hyperbolic groups. We prove that if a locally compact compactly generated group G admits one asymptotic cone that is a real tree and whose natural transitive isometric action is focal, then G must be a focal hyperbolic group. As an application, we characterize connected Lie groups and linear algebraic groups over an ultrametric local field of characteristic zero having cut-points in one asymptotic cone. We prove several results for locally compact lacunary hyperbolic groups, and extend the characterization of finitely generated lacunary hyperbolic groups to the setting of locally compact groups. We moreover answer a question of Olshanskii, Osin and Sapir about subgroups of lacunary hyperbolic groups.

math.GR↗

arXiv2016

Groups acting on trees with almost prescribed local action

We investigate a family of groups acting on a regular tree, defined by prescribing the local action almost everywhere. We study lattices in these groups and give examples of compactly generated simple groups of finite asymptotic dimension (actually one) not containing lattices. We also obtain examples of simple groups with simple lattices, and we prove the existence of (infinitely many) finitely generated simple groups of asymptotic dimension one. We also prove various properties of these groups, including the existence of a proper action on a CAT(0) cube complex.

math.GR↗

arXiv2015

The scale function and tidy subgroups

This exposition article arose from two talks given during the Oberwolfach Arbeitsgemeinschaft on Totally Disconnected Groups in October 2014. This is an introduction to the structure theory of totally disconnected locally compact groups initiated by Willis in 1994. The two main tools in this theory are the scale function and tidy subgroups, for which we present several properties and examples. As an illustration of this theory, we give a proof of the fact that the set of periodic elements in a totally disconnected locally compact group is always closed, and that such a group cannot have ergodic automorphisms as soon as it is non-compact.

math.GR↗

arXiv2014

The divergence of the special linear group over a function ring

We compute the divergence of the finitely generated group SLn(O_S), where S is a finite set of valuations of a function field, and O_S is the corresponding ring of S-integer points. As an application, we deduce that all its asymptotic cones are without cut-points.

math.GR↗