AEI-DFG: Mapping class groups from above and below

• Applicant: Dr. Elia Fioravanti
• Subject Area: Mathematics
• Term: since 2024
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 541703614

There are two fundamentally different approaches to analyzing the structure of an infinite, finitely generated group. One option is to observe the group from above: sitting at a basepoint at infinite distance and looking down on the group, so that it resembles a continuous object amenable to topological and geometric methods. This strategy, pioneered by Gromov in the 80s with the introduction of asymptotic cones, has become a prime tool in addressing questions related to quasi-isometric rigidity and other coarse geometric aspects of groups. Around the same time, similar ideas were also a major driving force behind Rips' theory of group actions on real trees, which ultimately led to Sela's breakthroughs in our understanding of the elementary theory of free and hyperbolic groups, as well as the structure of their automorphisms and homomorphisms. An alternative option is to observe the group from below: attempting to glean information on the group from the shadows that it projects in its quotients, particularly finite ones. This is the rationale behind the viciously difficult problem of profinite rigidity - whether a given group is determined by its profinite completion - which has started to see some initial progress only in recent years, despite having been put forth as far back as Grothendieck's work in the 70s. Our project will rely on both perspectives in order to study mapping class groups of finite-type surfaces. These are some of the most studied groups in mathematics because of their immensely rich and complicated structure, and also for their relevance to numerous distinct areas of research: from low-dimensional topology to complex geometry, from group theory to algebraic topology. Yet many fundamental questions about mapping class groups remain open: Is their elementary theory stable? Are they Kähler? Are they omnipotent? As our first goal, we will develop a version of Rips' theory for actions on higher-dimensional real trees, namely real cubings. Asymptotic cones of mapping class groups (as well as RAAGs and virtually special groups) are real cubings, which makes such a theory an important first step towards addressing model-theoretic aspects of mapping class groups. Secondly, we aim to make progress towards showing that mapping class groups are not Kähler. This will be achieved by exploring the isomorphism types of normal subgroups of both Kähler groups and mapping class groups. Finally, we will investigate quotients of mapping class groups and Kähler groups, with a special focus on those of Burnside type. We hope that this will shed light on various questions regarding residual properties of mapping class groups, particularly those on omnipotence and existence of non-elementary hyperbolic quotients. Our combined expertise on mapping class groups, real cubings, Kähler geometry, model theory and small cancellation techniques places us in prime position to make concrete and significant progress in each of these directions.

• DFG Programme: Research Grants
• International Connection: Spain
• Partner Organisation: Agencia Estatal de Investigación
• Co-Investigators: Professor Dr. Claudio Llosa Isenrich; Professorin Dr. Katrin Tent
• Cooperation Partners: Dr. Javier Aramayona; Dr. Montserrat Casals-Ruiz