Coarse medians as a hyperbolicity surrogate

• Applicant: Dr. Elia Fioravanti
• Subject Area: Mathematics
• Term: since 2023
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 515507199

Geometric group theory is the study of infinite, finitely generated groups through their actions on metric spaces. It is a new and dynamic field, thriving on its many connections to algebra, low-dimensional topology and differential geometry. A natural and particularly challenging problem in this setting has been understanding how the properties of a finitely generated group G will reflect onto those of its outer automorphism group Out(G). Already fixing G leads to beautiful and complicated theories, such as those of mapping class groups and Out(F_n). But what is even more remarkable is that automorphisms of arbitrary “negatively curved'' groups -- fundamental groups of negatively curved singular spaces -- can be studied in full generality, and display a rich structure that is intimately related to actions on R-trees and amalgamated-product splittings. Gaining a similar understanding of Out(G) when G is just “non-positively curved'' would have far-reaching implications, as many groups of a geometric origin only satisfy this weaker assumption. Unfortunately, none of the classical techniques carry over into this setting, and there is little indication that the picture should not be completely wild in this generality. Recent evidence suggests, however, that there is actually a great deal of structure regulating automorphisms of “cocompactly cubulated'' groups -- fundamental groups of non-positively curved spaces with a decomposition into cubes. Such groups are still very general and include all known non-positively curved groups with interesting automorphisms. The full picture is still waiting to be uncovered here and it has the potential to become a truly unifying approach, greatly generalising recent breakthroughs on automorphisms of right-angled Artin groups, while also further connecting them to the classical theory of Out(F_n) and mapping class groups. My proposed research will take the first steps in this exciting program. First, I will study Out(G) in this extreme level of generality, aiming to show that it is finitely generated for all cocompactly cubulated groups G. An important tool will be the action of Out(G) on the space of cubulations of G and on its natural coarsification: the space of coarse median structures. Second, I will investigate some of the finer questions on automorphisms in the slightly more restricted setting of special groups (in the Haglund-Wise sense) and right-angled Artin groups. I will demonstrate how the behaviour of fixed subgroups and growth rates, in particular, follows a pattern that is both restrained and surprisingly rich and variegated. Third, I will show that the class of cocompactly cubulated groups is even broader than expected, by developing new cubulating procedures taylored to non-hyperbolic groups. The fundamental new idea, throughout the project, is to make up for the lack of negative curvature by means of a “coarse median structure'', a concept recently introduced by Brian Bowditch.

• DFG Programme: Independent Junior Research Groups