Meeting on Arithmetic groups, Topology and Automorphic forms
IHES, Bures-sur-Yvette, June 16-18 2025

Arithmetic lattices are central to the theory of automorphic forms, whose main objects of study are functions with invariance properties with respect to arithmetic groups. The cohomology of these groups is thus a particular instance which is relatively easily accessible and has been the focus of much study over the past years. It also relates to their deformation theory.

More recently it has been realised that mapping class groups of surfaces admit many representations to certain arithmetic groups. These representations originally come from so-called TQFTs but are realised very concretely by matrix generators. Despite being well-studied the remain quite mysterious, in particular their rigidity properties are still not well-understood.

The aim of this short meeting is to bring together, on a mostly local scale, people interested in these two fields and see what interactions this may bring to light through expository talks on both topics and discussion sessions.

The meeting will take place at the Institut des hautes études scientifiques. Financial and admnistrative support is provided by the IHES, université Aix--Marseille and ANR through the AGDE grant. Talks will take place in the "Salle Alix et Marwan Lahoud".

Contact : Amina Abdurrahman (amina@ihes.fr), Léo Bénard (leo.BENARD@univ-amu.fr), Jean Raimbault (jean.raimbault@univ-amu.fr).

Participants

• Amina Abdurrahman
• Léo Bénard
• Pierre Charollois
• Lilou Domingues
• Sami Douba
• Alejandro Garcia
• Pierre Godfard
• Katia Hadj Said
• Veronica Fantini
• Julien Marché
• Arnaud Maret
• Gregor Masbaum
• Joan Porti
• Jean Raimbault
• Ramanujan Santharoubane
• Franco Vargas Pallete

Schedule

	Lundi	Mardi	Mercredi
9h30-10h30		Franco Vargas Pallete	Sami Douba
11h-12h		Pierre Charollois	Joan Porti
14h-15h		Gregor Masbaum
15h30-16h30	Arnaud Maret	Ramanujan Santharoubane
17h-18h	Veronica Fantini	Julien Marché (TBC)

Abstracts

Pierre Charollois

TBA: TBA

Sami Douba

Negative curvature, unipotents, and rational entries: We discuss various questions surrounding the relationship between negative/nonpositive curvature of finitely generated groups and the availability of faithful linear representations containing no nontrivial unipotents and consisting entirely of rational matrices.

Veronica Fantini

Resurgence and summability for perturbative invariants of knots: Motivated by complex Chern-Simons theory, a perturbative approach to quantum invariants of knots and 3-manifolds has been extensively studied. In a nutshell, new invariants have been defined in the form of divergent power series (e.g. Dimofte–Garoufalidis perturbative invariant of hyperbolic knots, the Ohtsuki series of homology 3-spheres, etc.), which should agree with the all-order asymptotics of the non-perturbative invariants (e.g. the Kashaev invariant, the Andersen–Kashaev state integral, the Witten–Reshetikhin–Turaev invariant, the Gukov–Pei–Putrov–Vafa invariant, etc.). In this talk, I will discuss the approach of resurgence and Borel-Laplace summability to the study of perturbative invariants of knots. These techniques are commonly used to study divergent power series and allow us to understand the relationship between perturbative and non-perturbative invariants.

Julien Marché

TBA: TBA

Arnaud Maret

Mapping class group quasi-moprhisms : from Entov--Polterovich to Gambaudo--Ghys: I will present on-going work with Vincent Humilière in which we're constructing quasi-morphisms for mapping class groups (of punctured spheres) from its action by Hamiltonian diffeomorphisms on character varieties. Our construction highlights potential striking similarities with a (very much different) construction by Gambaudo--Ghys.

Gregor Masbaum

Integral TQFT representations of mapping class groups: In joint work with Gilmer, we constructed an integral version of Witten-Reshetikhin-Turaev SO(3)-TQFT at roots of unity of prime order. Here "integral" means that the coefficients lie in a ring of cyclotomic integers. This leads to mapping class group representations into arithmetic groups with interesting properties. The aim of this talk is to explain how this works and maybe discuss some applications, if time allows.

Joan Porti

Character varieties in characteristic p: The variety of SL₂(C) characters of a finitely generated group is defined by polynomials with rational coefficients. In fact they are dyadic, eg lie in Z[1/2], so for a prime p>2 we may consider the reduction mod p of the variety of characters, and this yields the variety of characters in SL₂(F_p) for an algebraically closed field F_p of characteristic p. The goal of the talk is to provide examples of ramifications, and I have more questions than results.

Ramanujan Santharoubane

On kernel of quantum representations of mapping class groups: This talk will be centered around some finite dimensional complex representations of mapping class groups of surfaces. More precisely, we will discuss quantum representations of mapping class groups arising from Witten-Reshetikhin–Turaev Topological Quantum Fields Theories. These are projective unitary finite dimensional complex representations of mapping class groups indexed by an integer called level. When the level is a prime number, the image of the representation lands in the integral points of an algebraic group G and is Zariski dense in G. One important open problem is to know if this image has finite index in G(Z). As I will explain, for a fixed prime level p, knowing the kernel of the representation might help knowing if the image is arithmetic or not. I will explain a joint work with Renaud Detcherry where we can give some information about this kernel, more precisely we compute the two first terms of the so-called h-adic approximation of the representation (which is a sequence of finite groups approximation of the representation).

Franco Vargas Pallete

TBA: TBA