1. Spectral gaps of hyperbolic surfaces
1.i) Schottky surfaces
In joint work with Naud we extended our earlier results to a bounded frequency version of [B2, Conj. 2]. This is a breakthrough. It marked the first time that random hyperbolic surfaces were shown to have the optimal expected spectral gaps.
submitted for publication(odnośnik otworzy się w nowym oknie)1.ii) Near optimal spectral gaps for finite area surfaces (Annals of Math, to appear)
In a paper with my student Hide, we proved [B2, Conj. 1]. This occurred unexpectedly early in the project and marks a huge success.
I showed that this result can be built upon to show that every number between 0 and one quarter (inclusive) is a limit point of the set of first non-zero spectra of arithmetic surfaces. This answers a question of P. Sarnak (Chern lectures, Berkeley, 2022).
In `Strongly convergent unitary representations of limit groups', joint with Louder and Hide, we show that there are large genus compact arithmetic hyperbolic surfaces with asymptotically optimal spectral gaps.
1.iii) Random closed surfaces
Work was done on the finalization of the papers `The asymptotic statistics of random covering surfaces' and `A random cover of a compact hyperbolic surface has relative spectral gap 3/16- \epsilon'.
Extracted paper `Core surfaces' with Puder. This has been published in Geometriae Dedicata.
The paper `A random cover of a compact hyperbolic surface has relative spectral gap 3/16- \epsilon' with Naud and Puder has been published in GAFA.
In 2025, with Puder and van Handel we proved one of the main aims of the project:
uniformly random degree n covering spaces of a closed hyperbolic surface have asymptotically optimal spectral gap with high probability. The article is submitted for publication.
This article catalyzed three further breakthroughs: polynomial rate of convergence of first eigenvalue in random degree n covers (Hide-Macera-Thomas) and the corresponding result in variable negative curvature (Hide-Moy-Naud).
Slightly later, the tools we developed were used to prove the corresponding result in the Weil-Petersson random model (reproving Anantharaman-Monk and obtaining a polynomial rate).
2. Random Unitary Representations of Surface Groups
The main result achieved here to date is that for random such representations, as the dimension of the unitary group tends to infinity, these representations converge weakly to the regular representation of the surface group. Here weakly means that if we fix any element of the group algebra surface group, the normalized traces of the element w.r.t. the finite-dimensional unitary representations converge to the normalized trace in the regular representations. This is the analog of a result of Voiculescu (Invent. math. 1991) about random unitary representations of free groups.
This result is achieved in a series of two papers:
published in Communications in Mathematical Physics(odnośnik otworzy się w nowym oknie)Geometry and Topology, to appear(odnośnik otworzy się w nowym oknie)3. Strong convergence of finite dimensional unitary representations and purely matricial field groups
My breakthrough with Hide made it clear that whether or not a discrete group has a sequence of finite dimensional unitary representations that strongly converge to the regular representation has strong implications for the spectral geometry of locally symmetric spaces with that fundamental group.
Following seminar talks at Princeton University, and important meetings and conversations at IAS Princeton, the following definition arose:
a discrete group is purely matricial field (PMF) if it has a sequence of finite dimensional unitary representations that strongly converge to the regular representation.
With Louder (2022) we proved that all limit groups are PMF.
With Thomas (2023) we established that all:
- Right-angled Artin groups
- Coxeter groups
- Hyperbolic 3 manifold groups
- and `most' arithmetic hyperbolic 4-and-up-manifold groups are PMF.
In joint work with M. de la Salle we have established a boundary of this property: SL_4(Z) is not PMF (to appear, Comptes Rendus Mathematique). It seems like the result obtained so far have opened up a very interesting and important avenue of investigation.
4. Miscellaneous
In joint work with PDRA Thomas and Yufei Zhao we produced a paper `Quantum Unique Ergodicity for Cayley graphs of Quasirandom groups' (Comm. Math. Phys.). While not entirely inline with the proposed research, it sheds light on the role that quasirandomness plays in the spectral theory of Cayley graphs (a phenomenon tracing back to work on spectral gap by Sarnak and Xue). The paper is to appear in Communications in Mathematical Physics.
In joint work with PDRA Calderon we produced a paper `Spectral gap for Schottky subgroups of SL2(Z)' that improves on Gamburds thesis (2002) for Schottky groups. While this paper is not about optimal spectral gap, it makes an important tehcnical contribution to spectral gaps of Schottky surfaces. The paper is to appear in J.E.M.S.
