Related papers: Hyperconvex representations and exponential growth

We study the geometry of hyperconvex representations of surface groups in ${\rm PSL}(d,\mathbb{C})$ and their deformation spaces: We produce a natural holomorphic extension of the classical Ahlfors--Bers map to a product of Teichm\"uller…

It is well known that a strictly convex minimand admits at most one minimizer. We prove a partial converse: Let $X$ be a locally convex Hausdorff space and $f \colon X \mapsto \left( - \infty , \infty \right]$ a function with compact…

Let $\Gamma$ be a finite group, let $\theta$ be an involution of $\Gamma$, and let $\rho$ be an irreducible complex representation of $\Gamma$. We bound $\dim \rho^{\Gamma^{\theta}}$ in terms of the smallest dimension of a faithful…

This is an elementary introduction to the representation theory of finite semigroups. We illustrate the Clifford-Munn correspondence between the representations of a semigroup and the representations of its maximal subgroups. The emphasis…

Let $G$ be a real linear semisimple algebraic group without compact factors and $\Gamma$ a Zariski dense subgroup of $G$. In this paper, we use a probabilistic counting in order to study the asymptotic properties of $\Gamma$ acting on the…

Let G be a connected, real, semisimple Lie group contained in its complexification G_C, and let K be a maximal compact subgroup of G. We construct a K_C-G double coset domain in G_C, and we show that the action of G on the K-finite vectors…

For a countable discrete group {\Gamma} and a minimal {\Gamma}-space X, we study the notion of ({\Gamma}, X)-boundary, which is a natural generalization of the notion of topological {\Gamma}-boundary in the sense of Furstenberg. We give…

Let $ G $ be a real simple linear connected Lie group of real rank one. Then, $ X := G/K $ is a Riemannian symmetric space with strictly negative sectional curvature. By the classification of these spaces, $X$ is a real/complex/quaternionic…

We prove that the Gromov boundary of every hyperbolic group is homeomorphic to some Markov compactum. Our reasoning is based on constructing a sequence of covers of $\partial G$, which is quasi-$G$-invariant wrt. the ball $N$-type (defined…

The maximal normal subgroup growth type of a finitely generated group is $n^{\log n}$. Very little is known about groups with this type of growth. In particular, the following is a long standing problem: Let $\Gamma$ be a group and $\Delta$…

We prove a number of unconditional statistical results of the Hecke coefficients for unitary cuspidal representations of $\operatorname{GL}(2)$ over number fields. Using partial bounds on the size of the Hecke coefficients, instances of…

In the first part of our paper we analyze bisolutions and inverses of (non-autonomous) evolution equations. We are mostly interested in pseudo-unitary evolutions on Krein spaces, which naturally arise in linear Quantum Field Theory. We…

We re-examine some topics in representation theory of Lie algebras and Springer theory in a more general context, viewing the universal enveloping algebra as an example of the section ring of a quantization of a conical symplectic…

Let $k$ be a field, $\tilde{G}$ a connected reductive $k$-group, and $\Gamma$ a finite group. In a previous work, the authors defined what it means for a connected reductive $k$-group $G$ to be "parascopic" for $(\tilde{G},\Gamma)$.…

Consider a geometrically finite Kleinian group $G$ without parabolic or elliptic elements, with its Kleinian manifold $M=(\H^3\cup \Omega_G)/G$. Suppose that for each boundary component of $M$, either a maximal and connected measured…

Bounds on the log partition function are important in a variety of contexts, including approximate inference, model fitting, decision theory, and large deviations analysis. We introduce a new class of upper bounds on the log partition…

We study the Poisson-Furstenberg boundary of random walks on permutational wreath products. We give a sufficient condition for a group to admit a symmetric measure of finite first moment with non-trivial boundary, and show that this…

We introduce "representative generation," extending the theoretical framework for generation proposed by Kleinberg et al. (2024) and formalized by Li et al. (2024), to additionally address diversity and bias concerns in generative models.…

We revisit a ballistic deposition process introduced by Atar, Athreya and Kang. Let $\mathcal{G}=(V,E)$ be a finite connected graph. We choose independently and uniformly vertices in $\mathcal{G}$. If a vertex $x$ is chosen and the previous…

If $g\in G$ is a non-trivial element in a residually finite group, then there exists by definition a finite group $Q$ and a homomorphism $\varphi: G \to Q$ such that $\varphi(g) \neq e$. The residual finiteness growth $\text{RF}_G$ of a…