Injections of mapping class groups. (English) Zbl 1225.57001

The first main theorem of the paper shows that for all \(g \geq 2\) there exists \(h >g\) and an injective homomorphism \(\phi : \text{Mod}_g \rightarrow \text{Mod}_h\), where \(\text{Mod}_k\) denotes the mapping class group of a closed surface \(\Sigma_k\) of genus \(k\). These are examples of type-preserving homomorphisms, i.e. \(\phi\) maps pseudo-Anosov and reducible elements to pseudo-Anosov and reducible elements respectively. Further, for these examples, the authors obtain a \(\phi\)-equivariant isometric embedding of \(Teich(\Sigma_g)\) into \(Teich(\Sigma_h)\), where \(Teich\) denotes Teichmüller space. The techniques of proof are surprisingly elementary and clever.
For punctured surfaces, the authors construct even more interesting examples. They show that for all \(g \geq 2\) there exists \(h >g\) and an injective homomorphism \(\psi : \text{Mod}_{g,1} \rightarrow \text{Mod}_{h,1}\) such that \(\psi\) is not type-preserving. Here, \(\text{Mod}_{k,1}\) denotes the mapping class group of a surface \(\Sigma_{k,1}\) of genus \(k\) with one puncture. They further show that there is no \(\psi (\text{Mod}_{g,1})\)-invariant proper subset of \(\text{Teich}(\Sigma_{h,1})\) that is convex with respect to the Teichmüller metric. These examples therefore are very different in flavour from the first set of examples.
Reviewer: Mahan Mj (Howrah)


57M07 Topological methods in group theory
20F34 Fundamental groups and their automorphisms (group-theoretic aspects)
30F60 Teichmüller theory for Riemann surfaces
32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
57M60 Group actions on manifolds and cell complexes in low dimensions
57R50 Differential topological aspects of diffeomorphisms
Full Text: DOI arXiv


[1] J Behrstock, D Margalit, Curve complexes and finite index subgroups of mapping class groups, Geom. Dedicata 118 (2006) 71 · Zbl 1129.57023 · doi:10.1007/s10711-005-9022-3
[2] R W Bell, D Margalit, Braid groups and the co-Hopfian property, J. Algebra 303 (2006) 275 · Zbl 1110.20028 · doi:10.1016/j.jalgebra.2005.10.038
[3] L Bers, Fiber spaces over Teichmüller spaces, Acta. Math. 130 (1973) 89 · Zbl 0249.32014 · doi:10.1007/BF02392263
[4] J S Birman, Mapping class groups and their relationship to braid groups, Comm. Pure Appl. Math. 22 (1969) 213 · Zbl 0167.21503 · doi:10.1002/cpa.3160220206
[5] J S Birman, H M Hilden, On the mapping class groups of closed surfaces as covering spaces, Ann. of Math. Studies 66, Princeton Univ. Press (1971) 81 · Zbl 0217.48602
[6] J S Birman, H M Hilden, On isotopies of homeomorphisms of Riemann surfaces, Ann. of Math. \((2)\) 97 (1973) 424 · Zbl 0237.57001 · doi:10.2307/1970830
[7] O V Bogopol\(^{\prime}\)skiĭ, D V Puga, On the embedding of the outer automorphism group \(\mathrm{Out}(F_n)\) of a free group of rank \(n\) into the group \(\mathrm{Out}(F_m)\) for \(m> n\), Algebra Logika 41 (2002) 123, 253 · Zbl 1067.20042 · doi:10.1023/A:1015350112208
[8] R D Canary, D B A Epstein, P Green, Notes on notes of Thurston (editor D B A Epstein), London Math. Soc. Lecture Note Ser. 111, Cambridge Univ. Press (1987) 3 · Zbl 0612.57009
[9] N M Dunfield, W P Thurston, Finite covers of random \(3\)-manifolds, Invent. Math. 166 (2006) 457 · Zbl 1111.57013 · doi:10.1007/s00222-006-0001-6
[10] J Eells, L Lemaire, Selected topics in harmonic maps, CBMS Regional Conference Series in Math. 50, Conference Board of the Math. Sciences (1983) · Zbl 0515.58011
[11] B Farb, D Margalit, A primer on mapping class groups · Zbl 1245.57002
[12] P Hall, The Eulerian functions of a group, Quart. J. Math., Oxf. Ser. 7 (1936) 134 · Zbl 0014.10402
[13] W J Harvey, M Korkmaz, Homomorphisms from mapping class groups, Bull. London Math. Soc. 37 (2005) 275 · Zbl 1066.57020 · doi:10.1112/S0024609304003911
[14] J Hubbard, H Masur, Quadratic differentials and foliations, Acta Math. 142 (1979) 221 · Zbl 0415.30038 · doi:10.1007/BF02395062
[15] E Irmak, Superinjective simplicial maps of complexes of curves and injective homomorphisms of subgroups of mapping class groups, Topology 43 (2004) 513 · Zbl 1052.57024 · doi:10.1016/j.top.2003.03.002
[16] E Irmak, Superinjective simplicial maps of complexes of curves and injective homomorphisms of subgroups of mapping class groups. II, Topology Appl. 153 (2006) 1309 · Zbl 1100.57020 · doi:10.1016/j.topol.2005.04.001
[17] N V Ivanov, Automorphisms of Teichmüller modular groups (editor O Viro), Lecture Notes in Math. 1346, Springer (1988) 199 · Zbl 0657.57004
[18] N V Ivanov, Automorphism of complexes of curves and of Teichmüller spaces, Internat. Math. Res. Notices 1997 (1997) 651 · Zbl 0890.57018 · doi:10.1155/S1073792897000433
[19] N V Ivanov, J D McCarthy, On injective homomorphisms between Teichmüller modular groups. I, Invent. Math. 135 (1999) 425 · Zbl 0978.57014 · doi:10.1007/s002220050292
[20] B J., H Masur, Y Minsky, Asymptotics of Weil-Petersson geodesics I: ending laminations, recurrence, and flows · Zbl 1216.32007 · doi:10.1007/s00039-009-0034-2
[21] R P Kent IV, C J Leininger, Subgroups of mapping class groups from the geometrical viewpoint (editors D Canary, J Gilman, J Heinonen, H Masur), Contemp. Math. 432, Amer. Math. Soc. (2007) 119 · Zbl 1140.30017
[22] S P Kerckhoff, The Nielsen realization problem, Ann. of Math. \((2)\) 117 (1983) 235 · Zbl 0528.57008 · doi:10.2307/2007076
[23] I Kra, On the Nielsen-Thurston-Bers type of some self-maps of Riemann surfaces, Acta Math. 146 (1981) 231 · Zbl 0477.32024 · doi:10.1007/BF02392465
[24] C Maclachlan, W J Harvey, On mapping class groups and Teichmüller spaces, Proc. London Math. Soc. \((3)\) 30 (1975) 496 · Zbl 0303.32020 · doi:10.1112/plms/s3-30.4.496
[25] H Masur, Interval exchange transformations and measured foliations, Ann. of Math. \((2)\) 115 (1982) 169 · Zbl 0497.28012 · doi:10.2307/1971341
[26] J D McCarthy, Automorphisms of surface mapping class groups. A recent theorem of N Ivanov, Invent. Math. 84 (1986) 49 · Zbl 0594.57007 · doi:10.1007/BF01388732
[27] J D McCarthy, A Papadopoulos, Dynamics on Thurston’s sphere of projective measured foliations, Comment. Math. Helv. 64 (1989) 133 · Zbl 0681.57002 · doi:10.1007/BF02564666
[28] G Mess, Unit tangent bundle subgroups of the mapping class group, Preprint IHES/M/90/30
[29] L Paris, D Rolfsen, Geometric subgroups of mapping class groups, J. Reine Angew. Math. 521 (2000) 47 · Zbl 1007.57014 · doi:10.1515/crll.2000.030
[30] K J Shackleton, Combinatorial rigidity in curve complexes and mapping class groups, Pacific J. Math. 230 (2007) 217 · Zbl 1165.57017 · doi:10.2140/pjm.2007.230.217
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.