We show that if is a discrete subgroup of the group of the isometries of , and if is a representation of into the group of the isometries of , then any -equivariant map 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 -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 is of divergence type and is non-elementary, then there exists a measurable map conjugating the actions of and . Related applications are discussed.
On montre que pour chaque groupe discrète d’isométries de l’espace hyperbolique de dimension , chaque représentation de dans le groupe Isom et pour chaque application -équivariante de en , il existe une extension de 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 . 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.
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
@article{AIF_2009__59_1_393_0, author = {Francaviglia, Stefano}, title = {Constructing equivariant maps for representations}, journal = {Annales de l'Institut Fourier}, pages = {393--428}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {59}, number = {1}, year = {2009}, doi = {10.5802/aif.2434}, zbl = {1171.57016}, mrnumber = {2514869}, language = {en}, url = {http://www.numdam.org/articles/10.5802/aif.2434/} }
TY - JOUR AU - Francaviglia, Stefano TI - Constructing equivariant maps for representations JO - Annales de l'Institut Fourier PY - 2009 SP - 393 EP - 428 VL - 59 IS - 1 PB - Association des Annales de l’institut Fourier UR - http://www.numdam.org/articles/10.5802/aif.2434/ DO - 10.5802/aif.2434 LA - en ID - AIF_2009__59_1_393_0 ER -
%0 Journal Article %A Francaviglia, Stefano %T Constructing equivariant maps for representations %J Annales de l'Institut Fourier %D 2009 %P 393-428 %V 59 %N 1 %I Association des Annales de l’institut Fourier %U http://www.numdam.org/articles/10.5802/aif.2434/ %R 10.5802/aif.2434 %G en %F AIF_2009__59_1_393_0
Francaviglia, Stefano. Constructing equivariant maps for representations. Annales de l'Institut Fourier, Volume 59 (2009) no. 1, pp. 393-428. doi : 10.5802/aif.2434. http://www.numdam.org/articles/10.5802/aif.2434/
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