Topology and Geometric Group Theory Seminar
A group is said to "virtually algebraically fiber" if it has a finite index subgroup admitting a map onto Z with finitely generated kernel. Stronger than finite generation, if the kernel is even of type F_n for some n then we say the group "virtually algebraically F_n-fibers". Right-angled Coxeter groups (RACGs) are a class of groups for which the question of virtual algebraic F_n-fibering is of great interest. In joint work with Eduard Schesler, we introduce a new probabilistic criterion for the defining flag complex that ensures a RACG virtually algebraically F_n-fibers. This expands on work of Jankiewicz--Norin--Wise, who developed a way of applying Bestvina--Brady Morse theory to the Davis complex of a RACG to deduce virtual algebraic fibering. We apply our criterion to the special case where the defining flag complex comes from a certain family of finite buildings, and establish virtual algebraic F_n-fibering for such RACGs. The bulk of the work involves proving that a "random" (in some sense) subcomplex of such a building is highly connected, which is interesting in its own right.