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Induced quasi-actions: a remark. (English) Zbl 1231.20039

From the introduction: In this paper we observe that the notion of an induced representation has an analog for quasi-actions and give some applications.
Let \(G\) be a group and \(\{X_i\}_{i\in I}\) be a finite collection of unbounded metric spaces.
Definition 1.1. A quasi-action \(G\overset\rho\curvearrowright\prod_iX_i\) preserves the product structure if each \(g\in G\) acts by a product of quasi-isometries, up to a uniformly bounded error. Note that we allow the quasi-isometries \(\rho(g)\) to permute the factors; i.e. \(\rho(g)\) is uniformly close to a map of the form \((x_i)\mapsto(\varphi_{\sigma^{-1}(i)}(x_{\sigma^{-1}(i)}))\) with a permutation \(\sigma\) of \(I\) and quasi-isometries \(\varphi_i\colon X_i\mapsto X_{\sigma(i)}\).
Associated to every quasi-action \(G\overset\rho\curvearrowright\prod_iX_i\) preserving product structure is the action \(G\overset\rho{_I}\curvearrowright I\) corresponding to the induced permutation of the factors; this is well-defined because the \(X_i\)’s are unbounded metric spaces. For each \(i\in I\), the stabilizer \(G_i\) of \(i\) with respect to \(\rho_I\) has a quasi-action \(G_i\curvearrowright X_i\) by restriction of \(\rho\). It is well-defined up to equivalence in the sense of B. Kleiner and B. Leeb [Commun. Anal. Geom. 9, No. 2, 239-260 (2001; Zbl 1035.53073), Definition 2.3].
If the permutation action \(\rho_I\) is transitive, all factors \(X_i\) are quasi-isometric to each other, and the restricted quasi-actions \(G_i\curvearrowright X_i\) are quasi-conjugate (when identifying different stabilizers \(G_i\) by inner automorphisms of \(G\)). The main result of this note is that in this case any of the quasi-actions \(G_i\curvearrowright X_i\) determines \(\rho\) up to quasi-conjugacy, and moreover any quasi-conjugacy class may arise as a restricted action.
Theorem 1.2. Let \(G\) be a group, \(H\) be a finite index subgroup, and \(H\overset\alpha\curvearrowright X\) be a quasi-action of \(H\) on an unbounded metric space \(X\). Then there exists a quasi-action \(G\overset\beta\curvearrowright\prod_{i\in G/H}X_i\) preserving product structure, where
(1) Each factor \(X_i\) is quasi-isometric to \(X\).
(2) The associated action \(G\overset\beta_{G/H}\curvearrowright G/H\) is the natural action by left multiplication.
(3) The restriction of \(\beta\) to a quasi-action of \(H\) on \(X_H\) is quasi-conjugate to \(H\overset\alpha\curvearrowright X\).
Furthermore, there is a unique such quasi-action \(\beta\) preserving the product structure, up to quasi-conjugacy by a product quasi-isometry. Finally, if \(\alpha\) is an isometric action, then the \(X_i\) may be taken isometric to \(X\) and \(\beta\) may be taken to be an isometric action.

MSC:

20F65 Geometric group theory
53C35 Differential geometry of symmetric spaces
57M07 Topological methods in group theory

Citations:

Zbl 1035.53073
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References:

[1] A. Reiter Ahlin, The large scale geometry of products of trees, Geom. Dedicata 92 (2002), 179 – 184. Dedicated to John Stallings on the occasion of his 65th birthday. · Zbl 1009.20033 · doi:10.1023/A:1019630514124
[2] M. Bonk, B. Kleiner, and S. Merenkov, Rigidity of Schottky sets, Amer. J. Math., to appear. · Zbl 1168.30005
[3] Marc Bourdon and Hervé Pajot, Rigidity of quasi-isometries for some hyperbolic buildings, Comment. Math. Helv. 75 (2000), no. 4, 701 – 736. · Zbl 0976.30011 · doi:10.1007/s000140050146
[4] Richard Chow, Groups quasi-isometric to complex hyperbolic space, Trans. Amer. Math. Soc. 348 (1996), no. 5, 1757 – 1769. · Zbl 0867.20033
[5] Andrew Casson and Douglas Jungreis, Convergence groups and Seifert fibered 3-manifolds, Invent. Math. 118 (1994), no. 3, 441 – 456. · Zbl 0840.57005 · doi:10.1007/BF01231540
[6] David Gabai, Convergence groups are Fuchsian groups, Ann. of Math. (2) 136 (1992), no. 3, 447 – 510. · Zbl 0785.57004 · doi:10.2307/2946597
[7] M. Gromov, Hyperbolic manifolds, groups and actions, Riemann surfaces and related topics: Proceedings of the 1978 Stony Brook Conference (State Univ. New York, Stony Brook, N.Y., 1978) Ann. of Math. Stud., vol. 97, Princeton Univ. Press, Princeton, N.J., 1981, pp. 183 – 213.
[8] A. Hinkkanen, Abelian and nondiscrete convergence groups on the circle, Trans. Amer. Math. Soc. 318 (1990), no. 1, 87 – 121. · Zbl 0699.30017
[9] Michael Kapovich, Bruce Kleiner, and Bernhard Leeb, Quasi-isometries and the de Rham decomposition, Topology 37 (1998), no. 6, 1193 – 1211. · Zbl 0954.53027 · doi:10.1016/S0040-9383(97)00091-8
[10] M. Kapovich, B. Kleiner, B. Leeb, and R. Schwartz, private communication.
[11] Bruce Kleiner and Bernhard Leeb, Rigidity of quasi-isometries for symmetric spaces and Euclidean buildings, Inst. Hautes Études Sci. Publ. Math. 86 (1997), 115 – 197 (1998). · Zbl 0910.53035
[12] Bruce Kleiner and Bernhard Leeb, Groups quasi-isometric to symmetric spaces, Comm. Anal. Geom. 9 (2001), no. 2, 239 – 260. · Zbl 1035.53073 · doi:10.4310/CAG.2001.v9.n2.a1
[13] Bernhard Leeb, A characterization of irreducible symmetric spaces and Euclidean buildings of higher rank by their asymptotic geometry, Bonner Mathematische Schriften [Bonn Mathematical Publications], vol. 326, Universität Bonn, Mathematisches Institut, Bonn, 2000. · Zbl 1005.53031
[14] Vladimir Markovic, Quasisymmetric groups, J. Amer. Math. Soc. 19 (2006), no. 3, 673 – 715. · Zbl 1096.20042
[15] G. D. Mostow, Strong rigidity of locally symmetric spaces, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1973. Annals of Mathematics Studies, No. 78. · Zbl 0265.53039
[16] Lee Mosher, Michah Sageev, and Kevin Whyte, Quasi-actions on trees. I. Bounded valence, Ann. of Math. (2) 158 (2003), no. 1, 115 – 164. · Zbl 1038.20016 · doi:10.4007/annals.2003.158.115
[17] Pierre Pansu, Métriques de Carnot-Carathéodory et quasiisométries des espaces symétriques de rang un, Ann. of Math. (2) 129 (1989), no. 1, 1 – 60 (French, with English summary). · Zbl 0678.53042 · doi:10.2307/1971484
[18] Dennis Sullivan, On the ergodic theory at infinity of an arbitrary discrete group of hyperbolic motions, Riemann surfaces and related topics: Proceedings of the 1978 Stony Brook Conference (State Univ. New York, Stony Brook, N.Y., 1978) Ann. of Math. Stud., vol. 97, Princeton Univ. Press, Princeton, N.J., 1981, pp. 465 – 496.
[19] Pekka Tukia, On quasiconformal groups, J. Analyse Math. 46 (1986), 318 – 346. · Zbl 0603.30026 · doi:10.1007/BF02796595
[20] Xiangdong Xie, Quasi-isometric rigidity of Fuchsian buildings, Topology 45 (2006), no. 1, 101 – 169. · Zbl 1083.51008 · doi:10.1016/j.top.2005.06.005
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