Automorphisms of Smooth Fine Curve Graphs
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Author(s)
Booth, Katherine Williams
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Abstract
In this work, we consider the automorphisms of fine curve graphs where the vertices are restricted to continuously k-differentiable curves. We show that for closed surfaces with genus at least 2, they are all induced by homeomorphisms of the surface. Specifically, we show that the automorphisms of these graphs are naturally isomorphic to Homeo^k(S), the subgroup of homeomorphisms that preserve the set of one-dimensional C^k submanifolds of the surface. This result extends analogous work of Farb--Margalit and Long--Margalit--Pham--Verberne--Yao.
In addition, we study in further depth the particular case when k=1 and provide local conditions that characterize Homeo^1(S). We show that there exists a collection of conditions that are both necessary and sufficient for a homeomorphism of the surface to be an element of this group. These conditions primarily depend upon the structure of the induced map on the projective tangent bundle.
Finally, we provide examples of several types of elements of Homeo^k(S) that are not diffeomorphisms. These include elements inducing discontinuous maps on the projective tangent bundle and having infinitely many non-differentiable points.
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Date
2025-03-31
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Dissertation