Conjugacy classes of big mapping class groups
Jes\'us Hern\'andez Hern\'andez, Michael Hru\v{s}\'ak, Israel Morales,, Anja Randecker, Manuel Sedano, Ferr\'an Valdez

TL;DR
This paper investigates the conjugacy classes of big mapping class groups of infinite-type surfaces, revealing conditions for the existence of dense and meager conjugacy classes using model-theoretic methods.
Contribution
It characterizes when the mapping class group has dense or meager conjugacy classes based on the surface’s topological features, a novel application of model theory.
Findings
All conjugacy classes are meager for every surface.
Existence of a somewhere dense conjugacy class depends on the surface's ends.
A dense conjugacy class exists if the surface has a unique maximal end.
Abstract
We describe the topological behavior of the conjugacy action of the mapping class group of an orientable infinite-type surface on itself. Our main results are: (1) All conjugacy classes of are meager for every , (2) has a somewhere dense conjugacy class if and only if has at most two maximal ends and no non-displaceable finite-type subsurfaces, (3) has a dense conjugacy class if and only if has a unique maximal end and no non-displaceable finite-type subsurfaces. Our techniques are based on model-theoretic methods developed by Kechris, Rosendal and Truss.
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Taxonomy
TopicsGeometric and Algebraic Topology · Homotopy and Cohomology in Algebraic Topology · Mathematical Dynamics and Fractals
