Sur le codage du flot géodésique dans un arbre

C⁎-algebras associated to graphs of groups

“…On the other hand, a GBS graph of groups (see Section 7) has infinite vertex groups, so its fundamental group is not discrete in Aut(T ). Since [13] establishes topological freeness for certain discrete subgroups of Aut(T ), including certain lattices containing torsion, the overlap of our topological freeness results with theirs is the case of graphs of trivial groups where the underlying graph Γ has finite Betti number greater than one. (Recall that the Betti number of Γ is the cardinality of the set of edges in the complement of any maximal subtree in Γ.)…”

“…Now a necessary condition for the existence of Patterson-Sullivan measure on Λ is that the critical exponent of Λ is finite and positive (see [13]); under our assumption that G is a graph of trivial groups, this condition holds if and only if the first Betti number of Γ is finite and greater than one. Free groups have no nontrivial elliptic elements, hence if Λ admits a Patterson-Sullivan measure then the zero measure condition of [13] is vacuously true. Note that Λ will be a lattice in Aut(T ) if and only if the underlying graph Γ is finite, in which case Λ is a uniform lattice.…”

“…We note that our Theorem 7.5 generalises their Lemma 20(iii) on topological freeness for Baumslag-Solitar groups. Topological freeness is also relevant in the paper [13] of BroiseAlamichel and Paulin. Their Remarque 4.2 gives a proof of topological freeness under several hypotheses.…”

Boundary 𝐶*-algebras for acylindrical groups

“…The K-groups of the boundary algebra A Γ are isomorphic to the Bowen-Franks invariants of flow equivalence for a certain subshift of finite type associated with the geodesic flow [C3]. This subshift was studied in [BP,6.3].…”

Dynamique sur le rayon modulaire et fractions continues en caractéristique p

“…Ceci implique en particulier que si M est le sous-groupe compact maximal de S, alors l'actionà droite de S/M sur Γ \ G/M est lâchement Bernoulli (donc mélangeante, mais ceciétait déjà connu, par le théorème de Howe-Moore [24], voir aussi [2]). Dans [4], nousétudions le cas général des réseaux des groupes algébriques semi-simples de rang 1 sur un corps local non archimédien, et montrons que l'on peut remplacer lâchement Bernoulli par Bernoulli (d'entropie finie). En fait, nous donnons des codages markoviens de flots géodésiques (discrets) sur des arbres munis d'actions très générales de groupes.…”