Algebraic Aspects of Mapping Class Groups and Related Groups – AlMaRe

The mapping class groups of Riemann surfaces play a predominant role in several fields of mathematics, including algebraic geometry, differential geometry and low-dimensional topology. By a classical result of Dehn, Nielsen and Baer, these groups can be identified with the outer automorphism groups of the fundamental groups of Riemann surfaces, which are quotients of free groups by a single relator. Thus the study of mapping class groups from the geometric and/or combinatorial viewpoints is very often intertwined with the study of the automorphism groups of free groups, and that of some remarkable subgroups of the latter such as braid groups and their generalizations. More recently, groups of 3-dimensional homology cylinders have proved to be very interesting “enlargements” of the mapping class groups.

This research project will focus on some algebraic aspects of mapping class groups of surfaces and all those related groups. Mainly, we will be interested in their homology, their residual properties, the problem of their generations/presentations as well as their linear representations. The distinguishing feature of our study will be the use of *central filtrations* on these groups to identify their algebraic structure. The study of central filtrations on the automorphism group of a free group started in the middle 60’s with the work of Andreadakis: he introduced what is now called the "Andreadakis filtration" on the subgroup IA of "Identity by Abelianization" automorphisms of a free group, and he started to compare this central filtration with the lower central series of the group IA. Central filtrations on mapping class groups were further studied by Johnson in the 80’s and, next, by Morita in the 90’s: they undertook a systematic study of the "Johnson filtration", which is the analogue of the Andreadakis filtration for the subgroup of the mapping class group acting trivially on the homology of the surface, namely the "Torelli group". A bit later, the subject of central filtrations opened up to algebraic geometry with the works of Hain and has revealed new enriching connections with other mathematical trends such as the theory of finite-type invariants of 3-manifolds with results of Garoufalidis & Levine, or geometric group theory by the works of Bartholdi & Grigorchuk. Despite the variety and deepness of the results obtained in the last fifty years, the algebraic structure of mapping class groups and related groups still remains very mysterious. Thus, we have identified for this research project three main objectives:

(1) Study the Andreadakis/Johnson filtrations (and its variants) on these groups;
(2) Compute the homology of these groups (stable and unstable, with twisted or ordinary coefficients);
(3) Study some new linear representations of these groups.

Of course, as they are stated here in full generality, those objectives are deemed to be difficult. Nonetheless we believe that any substantial progress in these directions would considerably reduce the mystery and, for that, we will carry out the following actions whose methods will oscillate between algebraic topology and quantum topology:

* Make an extensive use of the new universal finite-type invariants to study central filtrations on mapping class groups;
* Use the theory of polynomial functors, notably to study stable phenomena in homology;
* Appeal to categorification to correct the lack of faithfulness of certain linear representations;
* Take advantage of the rich interplay between several families of groups that are related to mapping class groups of surfaces.