Thurston boundary of Teichmüller spaces

and the commensurability modular group

Authors:
Indranil Biswas, Mahan Mitra and Subhashis Nag

Journal:
Conform. Geom. Dyn. **3** (1999), 50-66

MSC (1991):
Primary 32G15, 30F60, 57M10, 57M50

DOI:
https://doi.org/10.1090/S1088-4173-99-00036-3

Published electronically:
April 12, 1999

MathSciNet review:
1684039

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Abstract | References | Similar Articles | Additional Information

Abstract: If is an unramified covering map between two compact oriented surfaces of genus at least two, then it is proved that the embedding map, corresponding to , from the Teichmüller space , for , to actually extends to an embedding between the Thurston compactification of the two Teichmüller spaces. Using this result, an inductive limit of Thurston compactified Teichmüller spaces has been constructed, where the index for the inductive limit runs over all possible finite unramified coverings of a fixed compact oriented surface of genus at least two. This inductive limit contains the inductive limit of Teichmüller spaces, constructed by I. Biswas, S. Nag and D. Sullivan, Determinant bundles, Quillen metrics and Mumford isomorphisms over the Universal Commensurability Teichmüller Space, * Acta Mathematica*, 176 (1996), 145-169, as a subset. The universal commensurability modular group, which was constructed in the above mentioned article, has a natural action on the inductive limit of Teichmüller spaces. It is proved here that this action of the universal commensurability modular group extends continuously to the inductive limit of Thurston compactified Teichmüller spaces.

**[BNS]**I. Biswas, S. Nag and D. Sullivan, Determinant bundles, Quillen metrics and Mumford isomorphisms over the Universal Commensurability Teichmüller Space,*Acta Mathematica*, 176, (1996), 145-169. MR**97h:32030****[BN1]**I. Biswas and S. Nag, Weil-Petersson geometry and determinant bundles over inductive limits of moduli spaces,*Lipa's Legacy*(Ed. J.Dodziuk and L.Keen), 51-80,*Contemporary Math, vol. 211*, (1997), Amer. Math. Soc. CMP**98:03****[BN2]**I. Biswas and S. Nag, Jacobians of Riemann surfaces and the Sobolev space on the circle,*Mathematical Research Letters*, 5, (1998), 281-292. CMP**98:16****[BN3]**I. Biswas and S. Nag, Commensurability automorphism groups and infinite constructions in Teichmüller theory,*Comptes Rendus Acad. Sci. (Paris)*, 327, (1998), 35-40.**[BN4]**I. Biswas and S. Nag, Limit constructions over Riemann surfaces and their parameter spaces, and the commensurability group actions, Preprint 1998.**[Bo1]**F. Bonahon, The geometry of Teichmüller space via geodesic currents,*Inventiones Math,*92, 1988, 139-162. MR**90a:32025****[Bo2]**F. Bonahon, Earthquakes on Riemann surfaces and on measured geodesic laminations,*Trans. Amer. Math Soc.*, 330, 1992, 69-95. MR**92f:57021****[CB]**A. Casson and S. Bleiler,*Automorphisms of surfaces after Nielsen and Thurston*, Cambridge Univ. Press, 1988. MR**89k:57025****[DH]**A. Douady and J. Hubbard, On the density of Strebel differentials,*Invent. Math.*, 30, (1975), 175-179. MR**53:796****[FLP]**A. Fathi, F. Laudenbach and V. Poenaru,*Travaux de Thurston sur le surfaces*, Asterisque, vol 66-67, (1979).**[H1]**R. E. Heisey, Embedding piecewise linear -manifolds into . The Proceedings of the 1981 Topology Conference (Blacksburg, Va., 1981),*Topology Proc.*, 6, (1981), 317-328. MR**83k:57010****[H2]**R. E. Heisey, Manifolds modelled on the direct limit of lines,*Pacific Jour. Math.*, 102, (1982), 47-54. MR**84d:57009****[HM]**J. Hubbard and H. Masur, Quadratic differentials and foliations,*Acta Mathematica*, 142, (1979), 221-274. MR**80h:30047****[M1]**H. Masur, Interval exchange transformations and measured foliations,*Annals of Math.*, 115, (1982), 169-200. MR**83e:28012****[M2]**H. Masur, Measured foliations and handlebodies,*Ergodic Th. Dynam. Systems*, 6, (1986), 99-116. MR**87i:57011****[N1]**S. Nag,*The Complex Analytic Theory of Teichmüller Spaces*, Wiley-Interscience, New York, (1988). MR**89f:32040****[N2]**S. Nag, Mathematics in and out of string theory,*Proc. 37th Taniguchi Symposium ``Topology and Teichmüller Spaces'',*Finland 1995, (Ed. S. Kojima, et. al.), World Scientific, (1996). CMP**99:06****[NS]**S. Nag and D. Sullivan, Teichmüller theory and the universal period mapping via quantum calculus and the space on the circle,*Osaka J. Math.*, 32, (1995), 1-34. MR**96c:32023****[Od]**C. Odden, The virtual automorphism group of the fundamental group of a closed surface, Thesis, Duke University, 1997.**[PH]**R. Penner and J. Harer,*Combinatorics of train tracks,*Annals Math Studies, Princeton Univ. Press, 1992. MR**94b:57018****[Str]**K. Strebel,*Quadratic differentials*, Springer Verlag, Berlin. MR**86a:30072****[W]**M. Wolf, The Teichmüller theory of harmonic maps,*Jour. Diff. Geom.*, 29, 1989, 449-479. MR**90h:58023**

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Additional Information

**Indranil Biswas**

Affiliation:
School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay 400005, India

Email:
indranil@math.tifr.res.in

**Mahan Mitra**

Affiliation:
School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay 400005, India

**Subhashis Nag**

Affiliation:
School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay 400005, India

DOI:
https://doi.org/10.1090/S1088-4173-99-00036-3

Received by editor(s):
April 27, 1998

Received by editor(s) in revised form:
January 28, 1999

Published electronically:
April 12, 1999

Article copyright:
© Copyright 1999
American Mathematical Society