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 $p : Y \rightarrow X$ 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 $p$, from the Teichmüller space $\mathcal {T}(X)$, for $X$, to $\mathcal {T}(Y)$ 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.

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Additional Information

**Indranil Biswas**

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

MR Author ID:
340073

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

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