## Thurston boundary of Teichmüller spaces and the commensurability modular group

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- by Indranil Biswas, Mahan Mitra and Subhashis Nag
- Conform. Geom. Dyn.
**3**(1999), 50-66 - DOI: https://doi.org/10.1090/S1088-4173-99-00036-3
- Published electronically: April 12, 1999
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## 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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## Bibliographic 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
- © Copyright 1999 American Mathematical Society
- 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
- MathSciNet review: 1684039