On reduced lantern relations in mapping class groups

Abstract

Let S be a hyperbolic Riemann surface with a finite area. Let G be the covering group of S acting on the hyperbolic plane H. In this paper, the author studies some algebraic relations in the mapping class group of Ṡ for Ṡ = S\{a point}. The author shows that the only possible relations between products of two Dehn twists and products of mapping classes determined by two parabolic elements of G are the reduced lantern relations. As a consequence, a partial solution to a problem posed by J. D. McCarthy is obtained.

Keywords

Dehn twists / Simple closed geodesics / Lantern relation

References

[1] Ahlfors L V, Bers L. Riemann’s mapping theorem for variable metrics. Ann. Math., 1960, 72: 385-404

[2] Beardon A. The Geometry of Discrete Groups, 1983, New York, Heidelberg, Berlin: Springer-Verlag

[3] Bers L. Fiber spaces over Teichmüller spaces. Acta Math., 1973, 130: 89-126

[4] Bers L. An extremal problem for quasiconformal mappings and a theorem by Thurston. Acta Math., 1978, 141: 73-98

[5] Birman J S. Braids, Links and mapping class groups, 1974, Princeton: Princeton University Press

[6] Dehn, M., Papers on Group Theory and Topology, Springer-Verlag, New York, 1987.

[7] Hamidi-Tehrani H. Groups generated by positive multi-twists and the fake lantern problem. Algebr. Geom. Topo., 2002, 2: 1155-1178

[8] Farkas H M, Kra I. Riemann Surfaces, 1980, New York, Berlin: Springer-Verlag

[9] Ivanov N V, McCarthy J D. On injective homomorphisms between Teichmuller modular groups, I. Invent. Math., 1999, 135: 425-486

[10] Kra I. On the Nielsen-Thurston-Bers type of some self-maps of Riemann surfaces. Acta Math., 1981, 146: 231-270

[11] Johnson D. Homeomorphisms of a surface which act trivially on homology. Proc. Amer. Math. Soc., 1979, 75: 119-125

[12] Margalit D. A lantern lemma. Algebr. Geom. Topo., 2002, 2: 1179-1195

[13] Nag S. Non-geodesic discs embedded in Teichmüller spaces. Amer. J. Math., 1982, 104: 339-408

[14] Thurston W P. On the geometry and dynamics of diffeomorphisms of surfaces. Bull. Amer. Math. Soc., 1988, 19: 417-431

[15] Zhang C. Pseudo-Anosov maps and fixed points of boundary homeomorphisms compatible with a Fuchsian group. Osaka J. Math., 2009, 46: 783-798

[16] Zhang C. On products of pseudo-Anosov maps and Dehn twists of Riemann surfaces with punctures. J. Aust. Math. Soc., 2010, 88: 413-428

[17] Zhang C. Pseudo-Anosov mapping classes and their representations by products of two Dehn twists. Chin. Ann. Math. Ser. B, 2009, 30(3): 281-292

[18] Zhang C. Invariant Teichmüller disks under hyperbolic mapping classes. Hiroshima Math. J., 2012, 42: 169-187

[19] Zhang, C., Pseudo-Anosov maps and pairs of filling simple closed geodesics on Riemann surfaces II, Tokyo J. Math., 36, 2013.

[20] Zhang C. On the minimum of asymptotic translation lengths of point pushing pseudo-Anosov maps on one punctured Riemann Surfaces, 2013

[21] Zhang C. Point-pushing pseudo-Anosov mapping classes and their actions on the curve complex, 2013