Actions on products of $$\textrm{CAT}(-1)$$ spaces
Tóm tắt
We show that for X a proper
$$\textrm{CAT}(-1)$$
space there is a maximal open subset of the horofunction compactification of
$$X\times X$$
, with respect to the maximum metric, that compactifies the diagonal action of an infinite quasi-convex group of the isometries of X. We also consider the product action of two quasi-convex representations of an infinite hyperbolic group on the product of two different proper
$$\textrm{CAT}(-1)$$
spaces.
Tài liệu tham khảo
Ballmann, W.: Lectures on Spaces of Nonpositive Curvature, volume 25 of DMV Seminar. Birkhäuser Verlag, Basel (1995). (With an appendix by Misha Brin)
Ballmann, W., Gromov, M., Schroeder, V.: Manifolds of Nonpositive Curvature. Progress in Mathematics, vol. 61. Birkhäuser Boston, Inc., Boston (1985)
Bourdon, M.: Structure conforme au bord et flot géodésique d’un \({\rm CAT}(-1)\)-espace. Enseign. Math. (2) 41(1–2), 63–102 (1995)
Bridson, M.R., Haefliger, A.: Metric Spaces of Non-positive Curvature. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 319. Springer, Berlin (1999)
Buyalo, S., Schroeder, V.: Elements of Asymptotic Geometry. EMS Monographs in Mathematics, European Mathematical Society (EMS), Zürich (2007)
Coornaert, M.: Sur le domaine de discontinuité pour les groupes d’isométries d’un espace métrique hyperbolique. Rend. Sem. Fac. Sci. Univ. Cagliari 59(2), 185–195 (1989)
Frances, C.: Lorentzian Kleinian groups. Comment. Math. Helv. 80(4), 883–910 (2005)
Furman, A.: Coarse-geometric perspective on negatively curved manifolds and groups. In: Burger, M., Iozzi, A. (eds.) Rigidity in Dynamics and Geometry (Cambridge, 2000), pp. 149–166. Springer, Berlin (2002)
García, T.: Compactification of a diagonal action on the product of \(\rm CAT(-1)\) spaces. Rep. SCM 3(1), 27–38 (2017)
Geninska, S.: The limit sets of subgroups of lattices in \(PSL (2,\mathbb{R})^r\). Geom. Dedicata 182, 81–94 (2016)
Kapovich, M., Leeb, B.: Finsler bordifications of symmetric and certain locally symmetric spaces. Geom. Topol. 22(5), 2533–2646 (2018)
Kapovich, M., Leeb, B., Porti, J.: Dynamics on flag manifolds: domains of proper discontinuity and cocompactness. Geom. Topol. 22(1), 157–234 (2018)
Krat, S.A.: On pairs of metrics invariant under a cocompact action of a group. Electron. Res. Announc. Amer. Math. Soc. 7, 79–86 (2001)
Link, G.: Asymptotic geometry in products of Hadamard spaces with rank one isometries. Geom. Topol. 14(2), 1063–1094 (2010)
Nicholls, P.J.: The Ergodic Theory of Discrete Groups. London Mathematical Society Lecture Note Series, vol. 143. Cambridge University Press, Cambridge (1989)
Papadopoulos, A.: Metric Spaces, Convexity and Non-positive Curvature, volume 6 of IRMA Lectures in Mathematics and Theoretical Physics, 2nd edn. European Mathematical Society (EMS), Zürich (2014)
Papasoglu, P., Swenson, E.: Boundaries and JSJ decompositions of CAT(0)-groups. Geom. Funct. Anal. 19(2), 559–590 (2009)
Swenson, E.L.: Quasi-convex groups of isometries of negatively curved spaces. Topol. Appl. 110(1), 119–129 (2001). (Geometric topology and geometric group theory (Milwaukee, WI, 1997))