Survival Numbers of Groups and Graphs with Emphasis on Z^d and Diestel-Leader Graphs. - Buckles, Kevin. (2017)
Abstract: The survival number of a metric space is a newly created statistic. For a given space, it equals the minimum amount of convex regions necessary to bound another convex region. Survival number is initially defined in terms of tessellations but can also be defined as a covering property. In this dissertation, we examine simple, connected graphs equipped with the path metric and finitely generated groups equipped with word metrics. We show the survival number of $\mathbb{Z}^2$, depending on the generating set, equals 3 or 4; and we then equate finding the survival number of $\mathbb{Z}^d$ for any $d ≥ 2$ with an open problem in combinatorial geometry. Last, we compute survival numbers of Diestel-Leader graphs and lamplighter groups.; Thesis (Ph.D.)--Tufts University, 2015.; Submitted to the Dept. of Mathematics.; Advisor: Moon Duchin.; Committee: Moon Duchin, Larry Guth, Mauricio Gutierez, and Kim Ruane.; Keyword: Mathematics.