Seminar

A group is A-special (resp. C-special) if it embeds in a right angled Artin (resp. Coxeter) group. For a 3-manifold, a cover is special if its fundamental group satisfies one of the above conditions. Following work of Wise, the existence of special covers of finite-volume hyperbolic 3-manifolds played a key role in Agol’s resolution of Thurston’s virtual conjectures. The resolution is not effective, in that no procedure for constructing the special cover is given and there is no explicit formula for the bound on the degree of such a cover. Subsequent work has investigated the quantitative nature of special covers. Chu used the fact that the figure-8 complement is arithmetic to show that the figure-8 complement has a C-special cover of degree 20. Covers up to degree 20 are accessible to a (sufficiently patient) computer search, but recognizing a special manifold remains challenging. In this talk I will describe the low_index module by Culler, Dunfield, and Goerner, and work (with SJSU undergraduates Kevin Schmidt and Ivanna Rodriguez Ramirez) to use this module to hunt for the smallest special cover of the figure-8 complement using the Dehn complex of the knot diagram.