A Brief Introduction to Lamplighter Groups and Diestel-Leader Graphs
Speaker: Austin Gaffglione
Date/Time: Friday, November 12th, 2:00 pm
Abstract: In 1990, W. Woess proposed the following question: Is every “nice” infinite graph quasi-isometric to the Cayley graph of some finitely generated group? In 2001, R. Diestel and I.Leader conjectured that some of graphs in the family they defined were not quasi-isometric to the Cayley graph of any finitely generated group. This was later proven to be true in a 2012 paper by A. Eskin, D. Fisher, and K. Whyte. We will examine the general construction of a Lamplighter group and observe some of its algebraic properties. We will also develop the notion of a horocyclic product of two trees, which will define a subfamily of the Diestel-Leader graphs. In particular, we will focus on the graph DL(2) and show that it is isomorphic to the Cayley graph of the Lamplighter Group L2 with respect to a certain choice for the generating set.
Видео A Brief Introduction to Lamplighter Groups and Diestel-Leader Graphs канала USF GradMath
Date/Time: Friday, November 12th, 2:00 pm
Abstract: In 1990, W. Woess proposed the following question: Is every “nice” infinite graph quasi-isometric to the Cayley graph of some finitely generated group? In 2001, R. Diestel and I.Leader conjectured that some of graphs in the family they defined were not quasi-isometric to the Cayley graph of any finitely generated group. This was later proven to be true in a 2012 paper by A. Eskin, D. Fisher, and K. Whyte. We will examine the general construction of a Lamplighter group and observe some of its algebraic properties. We will also develop the notion of a horocyclic product of two trees, which will define a subfamily of the Diestel-Leader graphs. In particular, we will focus on the graph DL(2) and show that it is isomorphic to the Cayley graph of the Lamplighter Group L2 with respect to a certain choice for the generating set.
Видео A Brief Introduction to Lamplighter Groups and Diestel-Leader Graphs канала USF GradMath
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