Riesz operators and some spherical representations for hyperbolic groups

Riesz operators and $L^{p}$-boundary representations for hyperbolic groups

We investigate Lp-boundary representations of hyperbolic groups. We prove that such representations are irreducible if and only if the corresponding Riesz operators are injective.

Spherical functions and rapid decay for hyperbolic groups

We investigate properties of some spherical fonctions defined on hyperbolic groups using boundary representations on the Gromov boundary endowed with the Patterson-Sullivan measure class. We prove sharp decay estimates for spherical functions as well as spectral inequalities associated with boundary representations. This point of view on the boundary allows us to view the so-called \emph{property RD} (also called \emph{Haagerup's inequality}) as a particular case of a more general behavior of spherical functions on hyperbolic groups. In particular, we give a constructive proof using a boundary…

Proper locally spherical hypertopes of hyperbolic type

Given any irreducible Coxeter group $C$ of hyperbolic type with non-linear diagram and rank at least $4$, whose maximal parabolic subgroups are finite, we construct an infinite family of locally spherical regular hypertopes of hyperbolic type whose Coxeter diagram is the same as that of $C$.

Hyperbolic structures for Artin-Tits groups of spherical type

The goal of this mostly expository paper is to present several candidates for hyperbolic structures on irreducible Artin-Tits groups of spherical type and to elucidate some relations between them. Most constructions are algebraic analogues of previously known hyperbolic structures on Artin braid groups coming from natural actions of these groups on curve graphs and (modified) arc graphs of punctured disks.

双曲群与球面极小曲面

Hyperbolic groups and spherical minimal surfaces

本文证明了负曲率流形的球面体积等于其基本群作用下单位球面商空间超极限中面积最小化极小曲面的面积，结合度量流理论与无挠双曲群正则表示极限完成证明。

Acylindrical hyperbolicity and Artin-Tits groups of spherical type

We prove that, for any irreducible Artin-Tits group of spherical type $G$, the quotient of $G$ by its center is acylindrically hyperbolic. This is achieved by studying the additional length graph associated to the classical Garside structure on $G$, and constructing a specific element $x_G$ of $G/Z(G)$ whose action on the graph is loxodromic and WPD in the sense of Bestvina-Fujiwara; following Osin, this implies acylindrical hyperbolicity. Finally, we prove that "generic" elements of $G$ act loxodromically, where the word "generic" can be understood in either of the two common usages: as a res…