Distortion and Tits alternative in smooth mapping class groups. (English) Zbl 1429.57036
Let \(S\) be a surface of finite type and \(K \subset S\) be a closed subset. Furthermore let \(\text{Diff}(S,K)\) be the group of orientation preserving \(C^\infty\)-diffeomorphisms of \(S\) that leave \(K\) invariant (as set). The smooth mapping class group is then defined as the quotient \[\mathcal{M}^\infty (S,K) = \text{Diff}(S,K) / \text{Diff}_0(S,K)\] where \(\text{Diff}_0(S,K)\) denotes the identity component of \(\text{Diff}(S,K)\). Furthermore denote by \(\mathcal{PM}^\infty(S,K)\) the subgroup of the smooth mapping class group consisting of elements which fix \(K\) pointwise. This is called the pure mapping class group.
Given a group \(G\) and a finite symmetric (\(\mathcal{G}^{-1} = \mathcal{G}\)) subset \(\mathcal{G} \subset G\). Denote by \(l_{\mathcal{G}}(g)\) the word length of the element \(g \in \langle \mathcal{G} \rangle\), defined as the minimal integer \(n\) such that \(g\) can be expressed as a product \(g = s_1 \cdots s_n\) with elements \(s_i \in \mathcal{G}\). An element \(g \in G\) is called distorted if it has infinite order and there exists a finite symmetric set \(\mathcal{G} \subset G\) such that \(g \in \langle \mathcal{G} \rangle\) and \(\lim_{n \to \infty}\frac{l_{\mathcal{G}}(g^n)}{n} = 0\).
The article under review investigates the pure mapping class group where the set \(K\) is a Cantor set. The following result is given: “Let \(S\) be a closed surface and \(K\) be a closed subset of \(S\) which is a Cantor set. Then the elements of \(\mathcal{PM}^\infty(S,K)\) are undistorted in \(\mathcal{M}^\infty(S,K)\).”
Furthermore it is shown that given a closed surface \(S\) and \(0 < \lambda < \frac{1}{2}\), then there are no distorted elements in the group \(\mathcal{M}^\infty(S, C_\lambda)\), where \(C_\lambda\) is an embedding of the standard ternary Cantor set with parameter \(\lambda\) in \(S\).
Mapping class groups of finite type satisfy the Tits alternative. It is investigated whether a similar statement holds for the groups where \(K\) is a Cantor set. The following result is given: “Let \(\Gamma\) be a finitely generated subgroup of \(\mathcal{M}^\infty(S, C_\lambda)\), then one of the following holds:
Given a group \(G\) and a finite symmetric (\(\mathcal{G}^{-1} = \mathcal{G}\)) subset \(\mathcal{G} \subset G\). Denote by \(l_{\mathcal{G}}(g)\) the word length of the element \(g \in \langle \mathcal{G} \rangle\), defined as the minimal integer \(n\) such that \(g\) can be expressed as a product \(g = s_1 \cdots s_n\) with elements \(s_i \in \mathcal{G}\). An element \(g \in G\) is called distorted if it has infinite order and there exists a finite symmetric set \(\mathcal{G} \subset G\) such that \(g \in \langle \mathcal{G} \rangle\) and \(\lim_{n \to \infty}\frac{l_{\mathcal{G}}(g^n)}{n} = 0\).
The article under review investigates the pure mapping class group where the set \(K\) is a Cantor set. The following result is given: “Let \(S\) be a closed surface and \(K\) be a closed subset of \(S\) which is a Cantor set. Then the elements of \(\mathcal{PM}^\infty(S,K)\) are undistorted in \(\mathcal{M}^\infty(S,K)\).”
Furthermore it is shown that given a closed surface \(S\) and \(0 < \lambda < \frac{1}{2}\), then there are no distorted elements in the group \(\mathcal{M}^\infty(S, C_\lambda)\), where \(C_\lambda\) is an embedding of the standard ternary Cantor set with parameter \(\lambda\) in \(S\).
Mapping class groups of finite type satisfy the Tits alternative. It is investigated whether a similar statement holds for the groups where \(K\) is a Cantor set. The following result is given: “Let \(\Gamma\) be a finitely generated subgroup of \(\mathcal{M}^\infty(S, C_\lambda)\), then one of the following holds:
- (1)
- \(\Gamma\) contains a free subgroup on two generators.
- (2)
- \(\Gamma\) has a finite orbit, i.e.there exists \(p \in C_\lambda\) such that the set \(\Gamma(p) := \{g(p) \,|\, g \in \Gamma\}\) is finite.”
Reviewer: Loreno Heer (Zürich)
MSC:
| 57S25 | Groups acting on specific manifolds |
| 37C85 | Dynamics induced by group actions other than \(\mathbb{Z}\) and \(\mathbb{R}\), and \(\mathbb{C}\) |
| 57K20 | 2-dimensional topology (including mapping class groups of surfaces, Teichmüller theory, curve complexes, etc.) |
Keywords:
mapping class groups; Tits alternative; Thompson’s group; distortion; smooth mapping class groups; Cantor setsReferences:
| [1] | Alibegovi\'c, Emina, Translation lengths in \({\rm Out}(F_n)\), Geom. Dedicata, 92, 87-93 (2002) · Zbl 1041.20024 · doi:10.1023/A:1019695003668 |
| [2] | B Collin Bleak et al., Centralizers in R. Thompson’s group \(V_n\), arXiv preprint arXiv:1107.0672 (2011). |
| [3] | Bavard, Juliette, Hyperbolicit\'e du graphe des rayons et quasi-morphismes sur un gros groupe modulaire, Geom. Topol., 20, 1, 491-535 (2016) · Zbl 1362.37086 · doi:10.2140/gt.2016.20.491 |
| [4] | Brin, Matthew G., Higher dimensional Thompson groups, Geom. Dedicata, 108, 163-192 (2004) · Zbl 1136.20025 · doi:10.1007/s10711-004-8122-9 |
| [5] | Ca Danny Calegari, Blogpost: https://lamington.wordpress.com/2014/10/24/mapping-class-groups-the-next-generation/ |
| [6] | Cannon, J. W.; Floyd, W. J.; Parry, W. R., Introductory notes on Richard Thompson’s groups, Enseign. Math. (2), 42, 3-4, 215-256 (1996) · Zbl 0880.20027 |
| [7] | Cooper, Daryl, Automorphisms of free groups have finitely generated fixed point sets, J. Algebra, 111, 2, 453-456 (1987) · Zbl 0628.20029 · doi:10.1016/0021-8693(87)90229-8 |
| [8] | Horbez, Camille, A short proof of Handel and Mosher’s alternative for subgroups of Out \((F_N)\), Groups Geom. Dyn., 10, 2, 709-721 (2016) · Zbl 1346.20048 · doi:10.4171/GGD/361 |
| [9] | CH2 Camille Horbez, The Tits alternative for the automorphism group of a free product, arXiv preprint arXiv:1408.0546 (2014). |
| [10] | Fisher, David, Groups acting on manifolds: around the Zimmer program. Geometry, rigidity, and group actions, Chicago Lectures in Math., 72-157 (2011), Univ. Chicago Press, Chicago, IL · Zbl 1264.22012 · doi:10.7208/chicago/9780226237909.001.0001 |
| [11] | Franks, John; Handel, Michael, Distortion elements in group actions on surfaces, Duke Math. J., 131, 3, 441-468 (2006) · Zbl 1088.37009 · doi:10.1215/S0012-7094-06-13132-0 |
| [12] | Franks, John; Handel, Michael, Periodic points of Hamiltonian surface diffeomorphisms, Geom. Topol., 7, 713-756 (2003) · Zbl 1034.37028 · doi:10.2140/gt.2003.7.713 |
| [13] | FN Louis Funar and Yurii Neretin, Diffeomorphisms groups of Cantor sets and Thompson-type groups, arXiv preprint arXiv:1411.4855 (2014). · Zbl 1499.57017 |
| [14] | Farb, Benson; Lubotzky, Alexander; Minsky, Yair, Rank-1 phenomena for mapping class groups, Duke Math. J., 106, 3, 581-597 (2001) · Zbl 1025.20023 · doi:10.1215/S0012-7094-01-10636-4 |
| [15] | FLP A. Fathi, F. Laudenbach, and V. Poenaru, Travaux de Thurston sur les surfaces, Ast\'erisque vol. 66, SMF, Paris, France. · Zbl 0406.00016 |
| [16] | Ghys, \'Etienne, Groups acting on the circle, Enseign. Math. (2), 47, 3-4, 329-407 (2001) · Zbl 1044.37033 |
| [17] | Gromov, M., Asymptotic invariants of infinite groups. Geometric group theory, Vol.2, Sussex, 1991, London Math. Soc. Lecture Note Ser. 182, 1-295 (1993), Cambridge Univ. Press, Cambridge |
| [18] | de la Harpe, Pierre, Topics in geometric group theory, Chicago Lectures in Mathematics, vi+310 pp. (2000), University of Chicago Press, Chicago, IL · Zbl 0965.20025 |
| [19] | Hirsch, Morris W., Differential topology, Graduate Texts in Mathematics 33, x+222 pp. (1994), Springer-Verlag, New York |
| [20] | Hmili, Hadda; Liousse, Isabelle, Dynamique des \'echanges d’intervalles des groupes de Higman-Thompson \(V_{r,m} \), Ann. Inst. Fourier (Grenoble), 64, 4, 1477-1491 (2014) · Zbl 1328.37037 |
| [21] | McCarthy, John, A “Tits-alternative” for subgroups of surface mapping class groups, Trans. Amer. Math. Soc., 291, 2, 583-612 (1985) · Zbl 0579.57006 · doi:10.2307/2000100 |
| [22] | Margulis, Gregory, Free subgroups of the homeomorphism group of the circle, C. R. Acad. Sci. Paris S\'er. I Math., 331, 9, 669-674 (2000) · Zbl 0983.37029 · doi:10.1016/S0764-4442(00)01694-3 |
| [23] | Milnor, John, Lectures on the \(h\)-cobordism theorem, Notes by L. Siebenmann and J. Sondow, v+116 pp. (1965), Princeton University Press, Princeton, N.J. · Zbl 0161.20302 |
| [24] | Navas, Andr\'es, Groups of circle diffeomorphisms, Chicago Lectures in Mathematics, xviii+290 pp. (2011), University of Chicago Press, Chicago, IL · Zbl 1236.37002 · doi:10.7208/chicago/9780226569505.001.0001 |
| [25] | Salazar-D\'\i az, Olga Patricia, Thompson’s group \(V\) from a dynamical viewpoint, Internat. J. Algebra Comput., 20, 1, 39-70 (2010) · Zbl 1266.20055 · doi:10.1142/S0218196710005534 |
| [26] | Zimmer, Robert J., Ergodic theory and semisimple groups, Monographs in Mathematics 81, x+209 pp. (1984), Birkh\"auser Verlag, Basel · Zbl 0571.58015 · doi:10.1007/978-1-4684-9488-4 |
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