Kent, Richard P. IV; Leininger, Christopher J. Uniform convergence in the mapping class group. (English) Zbl 1153.57013 Ergodic Theory Dyn. Syst. 28, No. 4, 1177-1195 (2008). In conversation at the 2005 Ahlfors-Bers colloquium, Ed Taylor asked the authors whether there is a formulation of convex cocompactness for the mapping class group, analogous to the following notion for Kleinian groups: a non-elementary Kleinian group \(\Gamma\) is convex cocompact if and only if the action of \(\Gamma\) on the limit set \(\Lambda_{\Gamma}\) is a uniform convergence action. Recall that an action of a group \(G\) on a perfect compact metrizable space \(X\) is a (discrete) convergence action if the diagonal action on the space of distinct triples in \(X\) is properly discontinuous, and that it is uniform if this associated action is cocompact. Theorem 1.3. Let \(G\) be a non-elementary subgroup of \(Mod(S).\) Then \(G\) is convex cocompact if and only if \(G\) acts as a uniform convergence group on \(Z\Lambda_{\Gamma}.\) Theorem 1.4. Suppose that \(G < Mod(S)\) is a non-elementary group. Then \(G\) is a convex cocompact if and only if \(\Omega_{G}\neq\varnothing\) and \(G\) acts cocompactly on it. Reviewer: V. V. Chueshev (Kemerovo) Cited in 5 Documents MSC: 57M50 General geometric structures on low-dimensional manifolds 57M60 Group actions on manifolds and cell complexes in low dimensions Keywords:convex cocompactness for mapping class group; Kleinian group; non-elementary group notion PDFBibTeX XMLCite \textit{R. P. Kent IV} and \textit{C. J. Leininger}, Ergodic Theory Dyn. Syst. 28, No. 4, 1177--1195 (2008; Zbl 1153.57013) Full Text: DOI arXiv References: [1] Masur, J. Anal. Math. 39 pp 1– (1981) [2] DOI: 10.2307/1971341 · Zbl 0497.28012 [3] Freden, Ann. Acad. Sci. Fenn. Ser. A I Math. 20 pp 333– (1995) [4] Ivanov, Subgroups of Teichm?ller Modular Groups (1992) [5] DOI: 10.2140/gt.2002.6.91 · Zbl 1021.20034 [6] Casson, Automorphisms of Surfaces After Nielsen and Thurston (1988) [7] Gehring, Proc. London Math. Soc.?(3) 55 pp 331– (1987) · Zbl 0628.30027 [8] DOI: 10.1353/ajm.2006.0003 · Zbl 1092.32008 [9] Gardiner, Complex Var. Theory Appl. 16 pp 209– (1991) · Zbl 0702.32019 [10] DOI: 10.1090/S0894-0347-98-00264-1 · Zbl 0906.20022 [11] Bowditch, Geometric Group Theory Down Under (Canberra, 1996) pp 23– (1999) [12] DOI: 10.1515/JGT.2007.054 · Zbl 1188.20041 [13] Tukia, New Zealand J. Math. 23 pp 157– (1994) [14] DOI: 10.2140/gt.2005.9.179 · Zbl 1082.30037 [15] DOI: 10.1007/BF02564666 · Zbl 0681.57002 [16] Masur, Ann. Acad. Sci. Fenn. Ser. A I Math. 20 pp 259– (1995) [17] DOI: 10.1007/PL00001643 · Zbl 0972.32011 [18] DOI: 10.1007/s002220050343 · Zbl 0941.32012 [19] DOI: 10.1215/S0012-7094-92-06613-0 · Zbl 0780.30032 [20] DOI: 10.2307/1971031 · Zbl 0322.32010 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.