Geometry & Topology, Vol. 8 (2004) Paper no. 36, pages 1301--1359.

On groups generated by two positive multi-twists: Teichmueller curves and Lehmer's number

Christopher J Leininger

Abstract. From a simple observation about a construction of Thurston, we derive several interesting facts about subgroups of the mapping class group generated by two positive multi-twists. In particular, we identify all configurations of curves for which the corresponding groups fail to be free, and show that a subset of these determine the same set of Teichmueller curves as the non-obtuse lattice triangles which were classified by Kenyon, Smillie, and Puchta. We also identify a pseudo-Anosov automorphism whose dilatation is Lehmer's number, and show that this is minimal for the groups under consideration. In addition, we describe a connection to work of McMullen on Coxeter groups and related work of Hironaka on a construction of an interesting class of fibered links.

Keywords. Coxeter, Dehn twist, Lehmer, pseudo-Anosov, mapping class group, Teichmueller

AMS subject classification. Primary: 57M07, 57M15. Secondary: 20H10, 57M25.

DOI: 10.2140/gt.2004.8.1301

E-print: arXiv:math.GT/0304163

Submitted to GT on 16 February 2004. (Revised 17 August 2004.) Paper accepted 11 October 2004. Paper published 19 October 2004.

Notes on file formats

Christopher J Leininger
Department of Mathematics, Columbia University
2990 Broadway MC 4448, New York, NY 10027, USA

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