Exotic mapping class group actions on the circle
Sang-hyun Kim, Thomas Koberda

TL;DR
This paper demonstrates that the standard faithful action of the mapping class group on the circle is not rigid, as finite index subgroups admit infinitely many non-conjugate faithful circle actions, answering a question by Farb.
Contribution
It shows the existence of infinitely many non-conjugate faithful circle actions for finite index subgroups of the mapping class group, revealing non-rigidity of the standard representation.
Findings
Existence of infinitely many non-conjugate faithful representations
Non-rigidity of the standard circle action of the mapping class group
Answers a question posed by Farb
Abstract
It has been known since the time of Nielsen that the mapping class group of a surface of genus and one puncture acts faithfully by homeomorphisms on the circle. In this note, we show that this standard representation of the mapping class group is not rigid, precisely, if is a finite index subgroup then there exist infinitely many non--conjugate faithful representations . We thus answer a question of B. Farb.
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Taxonomy
TopicsGeometric and Algebraic Topology · Mathematical Dynamics and Fractals · Homotopy and Cohomology in Algebraic Topology
