Action rigidity for free products of hyperbolic manifold groups
[Rigidité des actions des produits libres de groupes de variétés hyperboliques]
Stark, Emily R.1 ; Woodhouse, Daniel J.2
1 Wesleyan University Department of Mathematics 265 Church Street Middletown, CT 06459 (USA)
2 University of Oxford Mathematical Institute Andrew Wiles Building, Woodstock Road Oxford, OX2 6GG (United Kingdon)
Annales de l'Institut Fourier, Online first, 42 p.

Deux groupes ont la même géométrie de modèle s’ils agissent proprement et cocompactement par isométries sur un même espace métrique propre et géodésique. Nous étudions les produits libres de réseaux uniformes dans les groupes d’isométries d’espaces symétriques de rang 1 et prouvons que, dans chaque classe de quasi-isométries, les groupes résiduellement finis qui ont la même géométrie de modèle sont abstraitement commensurables. Notre résultat donne les premiers exemples de groupes hyperboliques qui sont quasi-isométriques mais qui n’ont virtuellement pas la même géométrie de modèle. Un élément important de la preuve est une généralisation d’un théorème de Leighton sur les revêtements finis communs des graphes. Le théorème principal utilise la notion de finitude résiduelle, et nous montrons que les extensions finies de réseaux uniformes des espaces symétriques de rang 1 qui ne sont pas résiduellement finis produisent des contre-exemples.

Two groups have a common model geometry if they act properly and cocompactly by isometries on the same proper geodesic metric space. We consider free products of uniform lattices in isometry groups of rank-1 symmetric spaces and prove, within each quasi-isometry class, that residually finite groups that have a common model geometry are abstractly commensurable. Our result gives the first examples of hyperbolic groups that are quasi-isometric but do not virtually have a common model geometry. An important component of the proof is a generalization of Leighton’s graph covering theorem. The main theorem depends on residual finiteness, and we show that finite extensions of uniform lattices in rank-1 symmetric spaces that are not residually finite would give counterexamples.

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DOI : 10.5802/aif.3585
Classification : 20F65, 20F67, 20E06, 57M07, 57M10
Keywords: Gromov Hyperbolicity, Residual finiteness, Bass–Serre theory, Action rigidity.
Mot clés : Hyperbolicité de Gromov, groupes résiduellement finis, théorie de Bass–Serre, rigidité des actions.
Stark, Emily R. 1 ; Woodhouse, Daniel J. 2

1 Wesleyan University Department of Mathematics 265 Church Street Middletown, CT 06459 (USA)
2 University of Oxford Mathematical Institute Andrew Wiles Building, Woodstock Road Oxford, OX2 6GG (United Kingdon) 
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Stark, Emily R.; Woodhouse, Daniel J. Action rigidity for free products of hyperbolic manifold groups. Annales de l'Institut Fourier, Online first, 42 p.

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