Constructing equivariant maps for representations

Francaviglia, Stefano

Annales de l'Institut Fourier, Tome 59 (2009) no. 1, pp. 393-428.

On montre que pour chaque groupe discrète d’isométries $G$ de l’espace hyperbolique de dimension $k$, chaque représentation $R$ de $G$ dans le groupe Isom$\left({ℍ}^n\right)$ et pour chaque application $R$-équivariante $F$ de ${ℍ}^k$ en ${ℍ}^n$, il existe une extension de $F$ dans le sens faible des mesures. On obtient donc, comme conséquence de ce fait, une extension d’un résultat de Besson, Courtois et Gallot sur l’existence d’une application équivariante qui n’augmente pas le volume. En plus, avec une hypothèse supplémentaire, on montre que notre extension faible est effectivement une vraie application mesurable du bord à l’infini de ${ℍ}^k$. On utilise alors ce résultat pour obtenir une version mesurable du résultat de Cannon et Thurston sur l’existence de courbes de Peano équivariantes. Enfin, on discute quelques applications.

We show that if $\Gamma$ is a discrete subgroup of the group of the isometries of ${ℍ}^k$, and if $\rho$ is a representation of $\Gamma$ into the group of the isometries of ${ℍ}^n$, then any $\rho$-equivariant map $F:{ℍ}^k\to {ℍ}^n$ extends to the boundary in a weak sense in the setting of Borel measures. As a consequence of this fact, we obtain an extension of a result of Besson, Courtois and Gallot about the existence of volume non-increasing, equivariant maps. Then, we show that the weak extension we obtain is actually a measurable $\rho$-equivariant map in the classical sense. We use this fact to obtain measurable versions of Cannon-Thurston-type results for equivariant Peano curves. For example, we prove that if $\Gamma$ is of divergence type and $\rho$ is non-elementary, then there exists a measurable map $D:\partial {ℍ}^k\to \partial {ℍ}^n$ conjugating the actions of $\Gamma$ and $\rho \left(\Gamma \right)$. Related applications are discussed.

Classification : 57M50, 37A99
Keywords: Hyperbolic spaces, discrete groups, isometries, representation, equivariant, barycenter, natural map, volume
Mot clés : espace hyperbolique, discrète group, isométries, représentation, équivariant, barycentre, application naturelle, volume

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