DISCRETE GROUPS FROM A COARSE PERSPECTIVE By JUSTIN I. SMITH A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY

of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy DISCRETE GROUPS FROM A COARSE PERSPECTIVE By Justin I. Smith May 2007 Chair: Alexander N. Dranishnikov Major: Mathematics We first use the functorial properties of coarse structures and coarse maps as well as algebraic properties to deduce that the asymptotic dimension of the Grigorchuk group is infinity. Next, by thinking in terms of coarse invariants rather than quasi-isometry invariants, we are able to extend the notion of a word metric to all countable groups. We describe nice relations between the asymptotic dimension of the countable group and the asymptotic dimensions of its finitely generated subgroups. Finally, using the so-called sublinear coarse structure, we are able to show that, for a large class of proper metric spaces including finitely generated groups, the asymptotic Assouad-Nagata dimension can be realized as the dimension of the sublinear Higson corona. Using this, we prove that crossing a space with the real numbers lifts the asymptotic Assouad-Nagata dimension by one.