Actions on products of \(\mathrm{CAT}(-1)\) spaces. (English) Zbl 1522.20171
In the paper under review, the authors consider the case of the product of two proper \(\mathrm{CAT}(-1)\) spaces, which is a \(\mathrm{CAT}(0)\) space (see [S. Geninska, Geom. Dedicata 182, 81–94 (2016; Zbl 1351.22007); G. Link, Geom. Topol. 14, No. 2, 1063–1094 (2010; Zbl 1273.20040)]). They show that for \(X\) a proper \(\mathrm{CAT}(-1)\) 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 \(X\). Furthermore, they consider the product action of two quasi-convex representations of an infinite hyperbolic group on the product of two different proper \(\mathrm{CAT}(-1)\) spaces.
Reviewer: Enrico Jabara (Venezia)
MSC:
| 20F65 | Geometric group theory |
| 20F67 | Hyperbolic groups and nonpositively curved groups |
| 57S30 | Discontinuous groups of transformations |
References:
| [1] | Ballmann, W., Lectures on Spaces of Nonpositive Curvature, volume 25 of DMV Seminar (1995), Basel: Birkh��user Verlag, Basel · Zbl 0834.53003 · doi:10.1007/978-3-0348-9240-7 |
| [2] | Ballmann, W.; Gromov, M.; Schroeder, V., Manifolds of Nonpositive Curvature. Progress in Mathematics (1985), Boston: Birkhäuser Boston, Inc., Boston · Zbl 0591.53001 · doi:10.1007/978-1-4684-9159-3 |
| [3] | Bourdon, M., Structure conforme au bord et flot géodésique d’un \({\rm CAT}(-1)\)-espace, Enseign. Math. (2), 41, 1-2, 63-102 (1995) · Zbl 0871.58069 |
| [4] | Bridson, MR; Haefliger, A., Metric Spaces of Non-positive Curvature. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] (1999), Berlin: Springer, Berlin · Zbl 0988.53001 |
| [5] | Buyalo, S.; Schroeder, V., Elements of Asymptotic Geometry (2007), Zürich: European Mathematical Society (EMS), Zürich · Zbl 1125.53036 · doi:10.4171/036 |
| [6] | 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) · Zbl 0799.53054 |
| [7] | Frances, C., Lorentzian Kleinian groups, Comment. Math. Helv., 80, 4, 883-910 (2005) · Zbl 1083.22007 · doi:10.4171/CMH/38 |
| [8] | Furman, A.; Burger, M.; Iozzi, A., Coarse-geometric perspective on negatively curved manifolds and groups, Rigidity in Dynamics and Geometry (Cambridge, 2000), 149-166 (2002), Berlin: Springer, Berlin · Zbl 1064.53025 · doi:10.1007/978-3-662-04743-9_7 |
| [9] | García, T., Compactification of a diagonal action on the product of \(\rm CAT(-1)\) spaces, Rep. SCM, 3, 1, 27-38 (2017) |
| [10] | Geninska, S., The limit sets of subgroups of lattices in \(PSL (2,\mathbb{R})^r\), Geom. Dedicata, 182, 81-94 (2016) · Zbl 1351.22007 · doi:10.1007/s10711-015-0129-x |
| [11] | Kapovich, M.; Leeb, B., Finsler bordifications of symmetric and certain locally symmetric spaces, Geom. Topol., 22, 5, 2533-2646 (2018) · Zbl 1417.53058 · doi:10.2140/gt.2018.22.2533 |
| [12] | Kapovich, M.; Leeb, B.; Porti, J., Dynamics on flag manifolds: domains of proper discontinuity and cocompactness, Geom. Topol., 22, 1, 157-234 (2018) · Zbl 1381.53090 · doi:10.2140/gt.2018.22.157 |
| [13] | Krat, SA, On pairs of metrics invariant under a cocompact action of a group, Electron. Res. Announc. Amer. Math. Soc., 7, 79-86 (2001) · Zbl 0983.51009 · doi:10.1090/S1079-6762-01-00097-X |
| [14] | Link, G., Asymptotic geometry in products of Hadamard spaces with rank one isometries, Geom. Topol., 14, 2, 1063-1094 (2010) · Zbl 1273.20040 · doi:10.2140/gt.2010.14.1063 |
| [15] | Nicholls, PJ, The Ergodic Theory of Discrete Groups (1989), Cambridge: Cambridge University Press, Cambridge · Zbl 0674.58001 · doi:10.1017/CBO9780511600678 |
| [16] | Papadopoulos, A., Metric Spaces, Convexity and Non-positive Curvature, volume 6 of IRMA Lectures in Mathematics and Theoretical Physics (2014), Zürich: European Mathematical Society (EMS), Zürich · Zbl 1296.53007 |
| [17] | Papasoglu, P.; Swenson, E., Boundaries and JSJ decompositions of CAT(0)-groups, Geom. Funct. Anal., 19, 2, 559-590 (2009) · Zbl 1226.20038 · doi:10.1007/s00039-009-0012-8 |
| [18] | Swenson, EL, Quasi-convex groups of isometries of negatively curved spaces, Topol. Appl., 110, 1, 119-129 (2001) · Zbl 0973.20037 · doi:10.1016/S0166-8641(99)00166-2 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
