Volume 8, issue 3 (2004)

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

Christopher J Leininger

Geometry & Topology 8 (2004) 1301–1359

arXiv: math.GT/0304163


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 Teichmüller 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.

Coxeter, Dehn twist, Lehmer, pseudo-Anosov, mapping class group, Teichmüller
Mathematical Subject Classification 2000
Primary: 57M07, 57M15
Secondary: 20H10, 57M25
Forward citations
Received: 16 February 2004
Revised: 17 August 2004
Accepted: 11 October 2004
Published: 19 October 2004
Proposed: Benson Farb
Seconded: Walter Neumann, Joan Birman
Christopher J Leininger
Department of Mathematics
Columbia University
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New York 10027