Cayley Graphs of the Integers with Infinite Generating Sets

Daniel Gulbrandsen
University of Wisconsin-Milwaukee
PhD Graduate Student – Teaching Assistant

“A question of Richard Schwartz asks: Are the metric spaces (Z,d_2) and (Z,d_3), where d_g denotes the word metric associated with the infinite generating set {g^n:n=0,1,2,…}U{-g^n:n=0,1,2,…}, quasi-isometric? In this talk we will address this question by investigating well known geometric invariants. We will also demonstrate certain families of maps that fail to be quasi-isometries between the two spaces in question. In particular, we will show that any polynomial with rational coefficients fails to be a quasi-isometry between these spaces.”