Barberis, Marco (2022) Rigidity properties of graphs associated with planar surfaces. PhD thesis, University of Warwick.

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Abstract

We prove two kinds of results on rigidity of graphs arising from punctured spheres. First, we prove that the graphs whose vertices are discs with a suitably bounded number of marked points, and whose edges are given by disjointness, are rigid, that is, every graph automorphisms is topologically induced by an extended mapping class. We will also extend this rigidity result to subgraphs of the curve graph, with similar bounds on marked points enclosed by each curve, obtaining a generalisation of Bowditch’s rigidity theorem for the strongly separating curve graph. Moreover, we will provide a complete topological classification of the rigid graphs of regions, which are graphs of isotopy classes of subsurfaces, sharpening a theorem of McLeay. Thus, our work verifies another case of Ivanov’s Metaconjecture, which states that sufficiently rich objects naturally associated with surfaces have the extended mapping class group as the group of their automorphisms.

The second group of results concerns the existence of exhaustions by finite rigid subgraphs, that is, such that every embedding into the ambient graph is induced by a global graph automorphism. We will study the case of the strongly separating curve graphs of both the seven-holed sphere and the eight-holed sphere, that is the graph whose vertices having at least three punctures in each complementary component, with edges given by disjointness. These are inspired by related work of Aramayona-Leininger on exhaustions of the regular curve graph. It follows that our graphs have the co-Hopfian property, that is every self-embedding of the entire graph is actually an automorphism.

Item Type:	Thesis (PhD)
Subjects:	Q Science > QA Mathematics
Library of Congress Subject Headings (LCSH):	Graph theory, Rigidity (Geometry), Surfaces, Curves
Official Date:	February 2022
Dates:	Date Event February 2022 UNSPECIFIED
Institution:	University of Warwick
Theses Department:	Mathematics Institute
Thesis Type:	PhD
Publication Status:	Unpublished
Supervisor(s)/Advisor:	Bowditch, B. H.
Sponsors:	Engineering and Physical Sciences Research Council
Extent:	xii, 168 pages : charts
Language:	eng

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