Dimension groups are invariants of orbital equivalence. We show in this lecture how to compute the dimension group of tree subshifts. Tree subshifts are defined in terms of extension graphs that describe the left and right extensions of factors of their languages: the extension graphs are trees. This class of subshifts includes classical families such as Sturmian, Arnoux-Rauzy subshifts, or else, codings of interval exchanges. We rely on return word properties for tree subshifts: every finite word in the language of a tree word admits exactly d return words, where d is the cardinality of the alphabet.
This is joint work with P. Cecchi, F. Dolce, F. Durand, J. Leroy, D. Perrin, S. Petite.[-]

Given a finite simple graph X, the right-angled Artin group associated to X is defined by the following very simple presentation: it has one generator per vertex of X, and the only relations consist in imposing that two generators corresponding to adjacent vertices commute. We investigate right-angled Artin groups from the point of view of measured group theory. Our main theorem is that two right-angled Artin groups with finite outer automorphism groups are measure equivalent if and only if they are isomorphic. On the other hand, right-angled Artin groups are never superrigid from this point of view: given any right-angled Artin group G, I will also describe two ways of producing groups that are measure equivalent to G but not commensurable to G.This is joint work with Jingyin Huang.[-]