Local and Asymptotic geometry of groups

Escalier, Amandine

Abstract

This manuscript presents the research work carried out during my thesis concerning LG-rigidity problems and orbit equivalence questions. After a short general introduction, we expose in the first part of this manuscript the results obtained on LG-rigidity and corresponding to the article [Esc20]. A vertex-transitive graph G is called Local-to-Global rigid if there exists R > 0 such that every othergraph whose balls of radius R are isometric to the balls of radius R in G is covered by G. An example of such a graph is given by the Bruhat-Tits building of PSLn(K) with n ≥ 4 and K a non-Archimedean local field of characteristic zero. In this part we extend this rigidity property to a class of graphs quasi-isometric to the building including torsion-free lattices of SLn(K).The proof is the occasion to prove a result on the local structure of the building. Weshow that if we fix a PSLn(K)-orbit in it, then a vertex is uniquely determined by the neighbouring vertices in this orbit.The second part presents (ongoing) work on orbit and measure equivalence. We say that two groups are orbit equivalent if they both admit an action on a same probability space that share the same orbits (up to a set of measure zero). In particular the Ornstein-Weiss theorem implies that all infinite amenable groups are orbit equivalent to the group of integers. Delabie, Koivisto, Le Maître and Tessera introduced a quantitative version of orbit equivalence and its measured couterpart to refine this notion between infinite amenable groups. They furthermore obtain obstructions to the existence of such equivalences using the isoperimetric profile.In this part we offer to answer the inverse problem (find a group being orbit or measure equivalent to a prescribed group with prescribed quantification) in the case of the group of integers or of the lamplighter group. To do so we use the diagonal products introduced by Brieussel and Zheng giving groups with prescribed isoperimetric profile.