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Local-Global Results On Discrete Structures, Alexander Lewis Stevens
Local-Global Results On Discrete Structures, Alexander Lewis Stevens
Electronic Theses and Dissertations
Local-global arguments, or those which glean global insights from local information, are central ideas in many areas of mathematics and computer science. For instance, in computer science a greedy algorithm makes locally optimal choices that are guaranteed to be consistent with a globally optimal solution. On the mathematical end, global information on Riemannian manifolds is often implied by (local) curvature lower bounds. Discrete notions of graph curvature have recently emerged, allowing ideas pioneered in Riemannian geometry to be extended to the discrete setting. Bakry- Émery curvature has been one such successful notion of curvature. In this thesis we use combinatorial …
Local Finiteness And Automorphism Groups Of Low Complexity Subshifts, Ronnie Pavlov, Scott Schmieding
Local Finiteness And Automorphism Groups Of Low Complexity Subshifts, Ronnie Pavlov, Scott Schmieding
Mathematics: Faculty Scholarship
We prove that for any transitive subshift X with word complexity function cn(X), if lim inf(log(cn(X)/n)/(log log log n)) = 0, then the quotient group Aut(X, σ)/〈 σ〉 of the automorphism group of X by the subgroup generated by the shift σ is locally finite. We prove that significantly weaker upper bounds on cn(X) imply the same conclusion if the gap conjecture from geometric group theory is true. Our proofs rely on a general upper bound for the number of automorphisms of X of range n in terms of word complexity, which may be …