The aim of this talk is to introduce Gromov-Hausdorff spaces introduced by É. Ghys in [1], which are spaces associated to Cayley graphs formed by (connected) subgraphs containing the identity element. They are compact spaces equipped with étale equivalence relations exhibiting interesting dynamics. In fact, if the group is not Z, the Gromov-Hausdorff space contains many closed saturated sets with different dynamics. We focus on the particular case of the abelian free group with two generators. We will also try to study if some of the properties of Cayley and Schreier graphs can be generalized to the Gromov-Hausdorff spaces such as the quasi-isometry, first introduced by M. Gromov in [2].

References:

[1] É. Ghys, Lamination par surfaces de Riemann. Panoramas & Synthèses, 8, (1999),

49-95.

[2] M. Gromov in M. Gromov, Asymtotic Invariants of Infinite Groups, in Geometric Group Theory, Vol.

2 (Sussex, 1991). London Mathematical Society Lecture Note Series 182, Cambridge

University Press, Cambridge (1993).