#### Volume 15, issue 3 (2011)

 Recent Issues
 The Journal About the Journal Editorial Board Editorial Interests Editorial Procedure Subscriptions Submission Guidelines Submission Page Policies for Authors Ethics Statement ISSN (electronic): 1364-0380 ISSN (print): 1465-3060 Author Index To Appear Other MSP Journals
Quantum traces for representations of surface groups in $\mathrm{SL}_2(\mathbb{C})$

### Francis Bonahon and Helen Wong

Geometry & Topology 15 (2011) 1569–1615
##### Abstract

We relate two different quantizations of the character variety consisting of all representations of surface groups in ${SL}_{2}$. One is the Kauffman skein algebra considered by Bullock, Frohman and Kania-Bartoszyńska, Przytycki and Sikora, and Turaev. The other is the quantum Teichmüller space introduced by Chekhov and Fock and by Kashaev. We construct a homomorphism from the skein algebra to the quantum Teichmüller space which, when restricted to the classical case, corresponds to the equivalence between these two algebras through trace functions.

##### Keywords
Kauffman skein relation, character variety, surface group, skein module, skein algebra, quantum Teichmüller theory
##### Mathematical Subject Classification 2010
Primary: 14D20, 57M25, 57R56
##### Publication
Revised: 10 January 2011
Accepted: 18 July 2011
Published: 19 September 2011
Proposed: Walter Neumann
Seconded: Jean-Pierre Otal, David Gabai
##### Authors
 Francis Bonahon Department of Mathematics University of Southern California 3620 S Vermont Ave, KAP 108 Los Angeles CA 90089-2532 USA http://www-rcf.usc.edu/~fbonahon/ Helen Wong Department of Mathematics Carleton College 1 North College St Northfield MN 55057 USA http://people.carleton.edu/~hwong