AMS@UCR

Abstract: In this talk we will discuss join work in progress with S. Lawton and F. Palesi on the (relative) \(\mathrm{SU}(2,1)\)–character variety for the once-holed torus. We consider the action of the mapping class group and describe a domain of discontinuity for this action, which strictly contains the set of convex-cocompact characters. We will also discuss the connection with the recent work of S. Schlich, and the inspiration behind this project, which lies in the rich theory developed for \(\mathrm{SL}(2,\mathbb{C})\)–character varieties by Bowditch, Minsky and others.

Abstract: Geometric structures on manifolds have been objects of interest since Thurston’s work and since then the work of Guichard and Wienhard has shown how certain representations of the fundamental groups of surfaces are closely related to geometric structures. Further objects of interest, introduced by Hitchin, are Higgs bundles and these are tied to surface group representations by the Nonabelian Hodge Correspondence. The goal of this talk is to highlight a method by which Higgs bundles can be used to determine geometric structures on related manifolds. \(\mathrm{SU}(2,1)\)-Higgs bundles satisfying a certain condition naturally give rise to developing maps for complex hyperbolic structures. These techniques also extend to a wider class of Higgs bundles, notably a family of \(\mathrm{SO}(2,n)\)-Higgs bundles. This is joint work between Brian Collier and myself.

Abstract: Given a surface, \(S\), of negative Euler characteristic and some maximal lamination, \(\lambda \subset S\), Bonahon developed shear-bend coordinates, which identify an open subset of the character variety, \(\chi(\pi_1(S),\mathrm{PSL}_2(\mathbb{C}))\), that contains all quasi-Fuchsian representations with an open subset of a finite dimensional -vector space, \(\mathcal{H}(\lambda,\mathbb{C})\). We show that at every Fuchsian representation, there is some definite radius depending only the choice of a train track (which yields a norm on ) and on the injectivity radius determined by the Fuchsian representation so that the ball of this radius in the shear-bend coordinates is entirely contained in the quasi-Fuchsian locus in \(\chi(\pi_1(S),\mathrm{PSL}_2(\mathbb{C}))\). This work is foreshadowed by the work of Epstein, Marden, and Markovich in which they prove a similar result in the special case that the bending cocycle is also a transverse measure. This work is paralleled by a work in progress which is hoped to provide a similar result in \(\chi(\pi_1(S),\mathrm{PSL}_d(\mathbb{C}))\) and yield definite balls that are contained in the Anosov locus.

Abstract: The moduli space \(M_g\) of genus \(g\) curves (or Riemann surfaces) is a central object of study in algebraic geometry. Its cohomology is important in many fields. For example, the cohomology of \(M_g\) is the same as the cohomology of the mapping class group, and is also related to spaces of modular forms. Using its properties as a moduli space, Mumford defined a distinguished subring of the cohomology of \(M_g\) called the tautological ring. The definition of the tautological ring was later extended to the compactification \(\bar{M}_g\) and the moduli spaces with marked points \(\bar{M}_{g,n}\). While the full cohomology ring of \(\bar{M}_{g,n}\) is quite mysterious, the tautological subring is relatively well understood, and conjecturally completely understood. In this talk, I’ll ask the question: which cohomology groups \(H^k(\bar{M}_{g,n})\) are tautological? And when they are not, how can we better understand them? This is joint work with Samir Canning and Sam Payne.

Abstract: Self-similar groups have emerged as a significant source of “exotic” examples in group theory, and they are intriguing objects to study. Random walks provide useful tools for understanding the structure of such groups, with applications to amenability problems, volume growth estimates, and more. In this talk, we will focus on iterated monodromy groups and discuss how the geometry of the associated limit set is related to the boundary behavior of random walks on the group. We will show that the Poisson boundary of symmetric finite range random walk is trivial when the limit set has Ahlfors-regular conformal dimension strictly less than 2. This is based on joint work with N. Matte Bon and V. Nekrashevych.

Abstract: Given a torsion-free word hyperbolic group \(\Gamma\) and a projective Anosov homomorphism \(\rho : \Gamma \to \mathrm{SL}(n,\mathbb{R})\), we construct a canonical (usually non-compact) locally homogeneous contact manifold \(\mathcal{M}_\rho\) equipped with an Axiom A flow \(\phi^t : \mathcal{M}_\rho \to \mathcal{M}_\rho\) whose holonomy is equal to \(\rho\). We will discuss several geometric and dynamical consequences: exponential mixing, existence of Ruelle-Pollicott resonant states, and a corresponding spectral gap. As an application, we obtain the exponential mixing of the geodesic flow associated to the Hilbert metric on a strictly convex real projective manifold. This is joint work with Daniel Monclair and Benjamin Delarue.

Abstract: We consider exponential mixing for volume preserving random dynamical systems on surfaces. Suppose that \((f_1,\dots,f_m)\) is a tuple of volume preserving diffeomorphisms of a closed surface \(M\). We now consider the uniform Bernoulli random dynamical system that this tuple generates on \(M\). We assume that this tuple satisfies a condition called being “expanding on average,” which means that there exist \(C,N > 0\) such that \(\mathbb{E}[\ln\lvert D f^N v\rvert]>C\) for all unit tangent vectors \(v\). From this assumption we show quenched exponential equidistribution as well as quenched exponential mixing. (This is joint work with Dmitry Dolgopyat.)

Abstract: We will discuss a correlation theorem for pairs of locally Hölder continuous potentials with strong entropy gaps at infinity on a topologically mixing countable Markov shift with the BIP property. This extends a result of Lalley on shifts of finite type, and we will explain its application to the dynamics of (pairs of) Hitchin representations of a punctured surface. This talk is based on joint work in progress with Lien-Yung Nyima Kao.

Abstract: In this talk we discuss the geometry of the hyperbolic locus in \(SL(2,\mathbb{R})^4\). The hyperbolic locus is the open set in \(SL(2,\mathbb{R})^n\), consisting of all n-tuples that would correspond to a uniformly hyperbolic set of matrices (of size n). In work by Avila, Bochi, and Yoccoz, the geometry of the hyperbolic locus is studied, and the geometry is explicitly defined in \(SL(2,\mathbb{R})^2\). Open questions were posed about the boundary of the hyperoblic locus in higher dimensions, and we address two of the questions here.

Abstract: We discuss an explicit geometric structures, i.e., \((G,X)\)-structures, interpretation of the \(G_2'\)-Hitchin component of a closed, oriented surface \(S\) of genus \(g \geq 2\). A definition of will be given as well as appropriate background and motivation for the problem. Then, using equivariant almost-complex curves in the pseudosphere \(\hat{\mathbb{S}}^{2,4}\), we describe the construction of the geometric structures for \(G_2'\)-Hitchin representations. Time permitting, we remark on how these structures are different than those of Guichard-Wienhard.

Abstract: Ever since Gromov showed the quasi isometries betwen hyperbolic spaces induce homeomorphisms on their visual boundaries, it has been common practice to use boundary of a metric space and use it to study the groups that act on the space isometrically. The problem with this method is that, once we leave the hyperbolic space, the visual boundary of a metric space may not be QI invariant any more. This motivated people to try to find good characterizations of hyperbolic-like directions in non-hyperbolic spaces. For example, people have studied the Morse boundary and contracting boundary of CAT(0) spaces. However, this task is particularly hard to accomplish in higher rank symmetric spaces. The existence of an isometrically embedded copy of \(\mathbb{R}^n\) for \(n \geq 2\) around every geodesic segment precludes any plausible characterizations for hyperbolicity being satisfied. Our goal is to develop a useful notion of sublinear Morse-ness of the action of a discrete subgroup \(\Gamma \subset Isom(X)\) on a higher rank symmetric space $X$ and show that this gives rise to a well defined, QI invariant boundary that has full measure with respect to the Patterson-Sullivan measure defined by \(\Gamma\).

Abstract: Thurston’s Hyperbolic Dehn Filling Theorem is a seminal result in the theory of \(3\)-manifolds. Given a single noncompact finite-volume hyperbolic \(3\)-manifold \(M\), the theorem provides a construction for a countably infinite family of closed hyperbolic 3-manifolds converging to \(M\) in a geometric sense. The theorem is a major source of examples of 3-manifolds admitting hyperbolic structures, and closely connects the topology of a 3-manifold \(M\) to the analysis of the character variety of its fundamental group in \(\mathrm{PSL}(2,\mathbb{C})\). In this talk, we discuss some analogs and generalizations of Thurston’s theorem in the context of general semisimple Lie groups. We will explain how our results provide a way to construct new examples of Anosov and relatively Anosov representations into higher-rank Lie groups; we will also discuss joint work with Jeff Danciger, which applies our results to construct exotic new examples of convex cocompact and geometrically finite groups acting on complex hyperbolic \(3\)-space.

Abstract: Let \(M\) be a hyperbolic 3-manifold. We say a sequence of distinct (non-commensurable) essential closed surfaces in \(M\) is asymptotically geodesic if their principal curvatures go uniformly to zero. When \(M\) is closed, these sequences exist abundantly by the Kahn-Markovic surface subgroup theorem, and we will discuss the fact that such surfaces are always asymptotically dense, even though they might not equidistribute. We will also talk about the fact that such sequences do not exist when \(M\) is convex-cocompact acylindrical of infinite volume. This joint work with Ben Lowe.

Abstract: We show that every positive Anosov representation of a free group into \(\mathrm{SO}(2n,2n-1)\) admits cocycles giving proper actions on affine \((4n-1)\)-space. We also exhibit fundamental domains for these actions. This is joint work with Jean-Philippe Burelle.

Abstract: The Laplacian on a compact Riemannian manifold is known to be diagonalizable, meaning there exists an orthonormal basis for \(L^2\) consisting of eigenfunctions of the operator. For decades, there has been significant interest in establishing upper bounds on the growth of \(L^p\) norms of these eigenfunctions in the high-frequency limit. These bounds have considerable implications for the theory of spectral multipliers and PDEs on Riemannian manifolds. They are closely related to the Stein-Tomas restriction theorem for the Fourier transform and provide insights into how an eigenfunction might concentrate its mass throughout the manifold.

A seminal result by Sogge in the 1980s provides \(L^p\) bounds on eigenfunctions that are “universal,” meaning they apply to any Riemannian manifold, regardless of its geometry. However, they are not typically the best possible ones. Indeed, proofs of the universal bounds generally rely on local properties of geodesics rather than their global dynamics. In recent years, it has been shown that the eigenfunctions on manifolds with nonpositive sectional curvature satisfy better bounds. In this talk, we will discuss these developments, with an emphasis on recent progress in joint works with Sogge and Huang concerning the case of critical exponents.