The Teichmüller distance between finite index subgroups of PSL<sub>2</sub>(<b>Z</b>)

Dragomir Saric

Knots in Washington XXV dedicated to Herbert Seifert on his 100 birthday. Conference on Knot Theory and its Ramifications
December 7-9, 2007
George Washington University
Washington, DC, USA

Organizers
Jozef H. Przytycki (GWU), Yongwu Rong (GWU), Alexander Shumakovitch (GWU), Dan Silver (U. South Al.), Hao Wu (GWU)

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The Teichmüller distance between finite index subgroups of PSL₂(Z)
by
Dragomir Saric
The City University of New York, Queens College
Coauthors: Vladimir Markovic

For a given ε > 0, we show that there exist two finite index subgroups of PSL₂(Z) which are (1+ε)-quasisymmetrically conjugated and the conjugation homeomorphism is not conformal. This implies that for any ε > 0 there are two finite regular covers of the Modular once punctured torus T₀ (or just the Modular torus) and a (1+ε)-quasiconformal between them that is not homotopic to a conformal map. As an application of the above results, we show that the orbit of the basepoint in the Teichmüller space T(S ) of the punctured solenoid S under the action of the corresponding Modular group has the closure in T(S ) strictly larger than the orbit and that the closure is necessarily uncountable. This is a joint work with V. Markovic.

Date received: November 23, 2007