Research interests

In general: geometric group theory and low-dimensional topology.

In particular: Let S be a connected surface with empty boundary; the mapping class group of S, denoted by MCG(S), is the group of isotopy classes of orientation preserving self-homeomorphisms of S.

My research is mainly centered in studying the local and large-scale geometry of MCG(S) (and its most famous extensions and subgroups) via its actions on several geometric objects (mainly connected graphs and other metric spaces).

If S is of finite type (finitely generated fundamental group), in particular my research is focused on the rigidity of said actions; with this I mean the following: Let X be a geometric object on which MCG(S) acts; more often than not, X is either a complex of curves (or multicurves, or arcs, etc.) or a Gromov-hyperbolic metric space. Are all automorphisms/isometries/quasi-isometries of X induced by a mapping class? If I have a graph morphism/isometric embedding/quasi-isometric embedding f, what conditions are sufficient for f to be (possibly a restriction of) an equivalence in the corresponding categories?

If S is of infinite type (fundamental group not finitely generated), besides studying the questions above, I am also interested in analogues of results in the finite-type setting, such as Is there a classification of the elements of MCG(S) analogue to Nielsen-Thurston's? Also, I am interested in results that are more native in this setting, taking advantage of the natural (and non-discrete) topology that MCG(S) has: In general, what topological-group-theoretic properties does MCG(S) have? These questions are highly non-trivial, and we have several partial answers.

In both settings, I have lately been interested in these subjects: embeddability of certain groups into MCG(S), description/stability of the cohomology of MCG(S), (if possible finite-type) models for Bredon cohomology for different families of subgroups, etc.

Due to my research preferences, I am also very interested in results concerning the local/global/large-scale geometry of the geometric objects on which MCG(S) acts, as well as more general geometric-group-theoretic results concerning (acylindrically/relatively) hyperbolic groups, right-angled Artin groups, etc.