Periodic Reporting for period 2 - UBIQGAP (The ubiquity of optimal spectral gaps)
Reporting period: 2022-07-01 to 2023-12-31
The presence of large spectral gaps is often useful for practical applications. Moreover, oftentimes there are pre-existing bounds on how large spectral gaps can be. Therefore there is a notion of optimal (or near-optimal) spectral gap, and these are highly desirable.
The proposal aims to address the question of whether the occurence of optimal or near optimal spectral gaps
are
1. possible?
2. common?
3. or in fact, typical?
for a variety of mathematical objects. In all cases, these mathematical objects have close parallels to physical systems.
The overall objectives are to answer the above questions about ubuiquity of optimal spectral gaps for
- hyperbolic surfaces of any type (compact, finite-area, or infinite area). Here, the spectral gap involves the spectrum of the Laplacian, or more complicated objects called resonances.
- hecke operators arising from unitary representations of residually finite discrete groups (where free groups already contain dififcult challenges).
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
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.
In forthcoming solo paper, I show 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 compact 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.
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
Geometry and Topology, to appear
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 bakc 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.
The focus of the project should now shift, so much as it is possible to obtain results, in the other directions outlined in the proposal B2.
Random compact surfaces
A major remaining open problem is whether in any natural model, random compact surfaces hae almost optimal spectral gaps. Work on this will continue in collaboration with team members and Puder.
Ramanujan surfaces
One of the most appealing remaining problems in the area of the grant proposal is whether there exist finite area Ramanujan hyperbolic surfaces of unbounded genus.
Progress on this problem will be attempted on lines of Quesiton 2, and Objectives 2 and 3 of B2.
Hecke operators
PI still plans, in collaboration with PDRAs, to carry out work on Broad Goal 3: Prove that generic unitary representations of fundamental groups of surfaces have
almost-optimal spectral gaps. This work should proceed via Objective 4 of B2.
Integration on representation varieties
The PI will continue the series of papers RURSG1 and RURSG2. This will constitute a full solution of Objective 6.
The PI has ongoing research on Broad Goal 3 that is expected to complete before end of project.
E. Cassidy (team member) has ongoing research here. He plans to show (amongst other things) that results of Puder and Parzanchevski extend to all stable characters of S_n.
This work will feed into major progress on Problem 1 (understanding of algebraic properties of Wilson loops).