Geometry, Topology and Dynamics Seminar
Peter B. Shalen
UIC
Margulis numbers and algebraic number fields
Abstract: This talk will discuss new developments in my project of relating
trace fields of hyperbolic 3-manifolds to their quantitative geometric
properties. I will focus on the following result, the proof of which
combines several different techniques:
Let $\alpha=1.32252\ldots$ denote the unique zero in $(1,\infty)$ of
the polynomial $x^8-x^5-x^3-3$. Let $K$ be any number field. Then up
to isometry there are at most finitely many closed, orientable
hyperbolic $3$-manifolds $M$ with the following properties:
(i) $M$ is non-Haken and $H_1(M;{\mathbb Z}_2)$ and $H_1(M;{\mathbb Z}_3)$ are trivial;
(ii) $M$ has trace field $K$; and
(iii) $\log\alpha=0.279\ldots$ is not a Margulis number for $M$.
Monday September 21, 2009 at 3:00 PM in SEO 612