arXiv subjects

Kevin Boucher

Publications and source records attributed to Kevin Boucher.

7 recordsLinked to original sources

arXiv2026

Uniformly affine actions on Banach spaces: growth of cocycles

We investigate growth properties of cocycles with values in uniformly bounded representations on super-reflexive Banach spaces; this includes $L^p$-spaces for $1<p<\infty$ as well as Hilbert spaces. We then study the generalized Hilbert compression of cocycles arising in this setting for the Property (T) groups $\mathrm{Sp}(n,1)$, $n\ge 2$, and establish the existence of uniformly Lipschitz affine actions with optimal growth.

math.GR↗

arXiv2024

Boundary representations of hyperbolic groups: the log-Sobolev case

We study boundary representations of hyperbolic groups $\Gamma$ on the (compactly embedded) function space $W^{\log,2}(\partial\Gamma)\subset L^2(\partial\Gamma)$, the domain of the logarithmic Laplacian on $\partial\Gamma$. We show that they are not uniformly bounded, and establish their exact growth (up a multiplicative constant): they grow with the square root of the length of $g\in\Gamma$. We also obtain $L^p$--analogue of this result. Our main tool is a logarithmic Sobolev inequality on bounded Ahlfors--David regular metric measure spaces.

math.GR↗

arXiv2024

Spectral gaps, critical exponents and representations of negatively curved groups

In this paper we introduce a notion of Poincar\'e exponent for isometric representations of discrete groups on Hilbert spaces. Similarly as growth exponents control the geometry this exponent is shown to control the size of spectral gaps. Following similar ideas as Patterson and Sullivan it is used in the case of negatively curved groups to construct weakly contained boundary representations reflecting the spectral properties of the original representation analogously as complementary series representations in the case of semi-simple Lie groups. This is exploited to deduced sharp estimates on spectral invariants. A quantitive property (T) \'a la Cowling is also established proving uniform bound on the mixing rate of representations of hyperbolic groups with property (T). Along the way some properties of boundary representations are discussed. A original characterisation of the positivity of the so-called Knapp-Stein operators and certain fusion rules on the boundary complementary series representations are established.

math.DS↗

arXiv2023

Sobolev spaces and uniform boundary representations

We prove uniform boundedness of certain boundary representations on appropriate fractional Sobolev spaces $W^{s,p}$ with $p>1$ for arbitrary Gromov hyperbolic groups. These are closed subspaces of $L^p$ and in particular Hilbert spaces in the case $p=2$. This construction allows us, for an appropriate choice of $p$, to approximate the trivial representation through uniformly bounded representations. This phenomenon does not have analogue in the setting of isometric representations whenever the hyperbolic group considered has the Property (T). The key is the introduction of a notion of metrically conformal operator on a metric space endowed with a conformal structure \`{a} la Mineyev and a metric analogue of the isomorphisms of Sobolev spaces induced by the Cayley transform.

math.GR↗

arXiv2020

Analogs of complementary series for CAT(-1) groups

In this paper we extend the construction of complementary series representations to convex-cocompact isometry groups of CAT(-1) spaces with conditionally negative metrics. Our approach is purely dynamical and generalizes the constructions known for negatively curved algebraic groups as $SO(n,1)$, $SU(n,1)$ or $SL_2(\mathbb{Q}_p)$ and their lattices to new examples as non-linear groups coming from lattices of certain hyperbolic buildings.

math.GR↗

arXiv2020

Special affine representations for hyperbolic groups

In this paper we extend the construction of special representations to Gromov hyperbolic groups which admits complementary series. We prove that these representations have a natural non-trivial reduced cohomology class $[c]$. An analogue of Kuhn-Vershik's formula is established and as a by-product a characterisation of hyperbolic groups that admit complementary series. Investigating dynamical properties of the cohomology class $[c]$ we prove an cocycle equidistribution theorem \'a la Roblin-Margulis and deduce the irreducibility of the associated affine actions. The irreducibility of the affine actions associated to the canonical class $[c]$ is original even in the case of uniform lattices in $SO(n,1)$, $SU(n,1)$ or $SL_2(\mathbb{Q}_p)$ with $n\ge 1$ and $p$ prime.

math.GR↗

arXiv2020

Random walks in negative curvature and boundary representations

In this paper we establish a version of the Margulis Roblin equidistribution theorem's for harmonic measures. As a consequence a von Neumann type theorem is obtained for boundary actions and the irreducibility of the associated quasi-regular representations is deduced.

math.DS↗