A group is said to "virtually algebraically fiber" if it has a finite index subgroup admitting a map onto \(\mathbb{Z}\) with finitely generated kernel. Stronger than finite generation, if the kernel is even of type \(\mathrm{F}_n\) for some \(n\) then we say the group "virtually algebraically \(\mathrm{F}_n\)-fibers". Right-angled Coxeter groups (RACGs) are a class of groups for which the question of virtual algebraic \(\mathrm{F}_n\)-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 \(\mathrm{F}_n\)-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 \(\mathrm{F}_n\)-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.

In the first half of the talk I will focus just on what Jankiewicz--Norin--Wise did, so in particular always \(n=1\), and then in the second half I will get into the \(n>1\) case and the specific examples.

Join Zoom meeting ID 978 1075 8446 or via the link below. Passcode: A group is called an ________ group if it admits an invariant mean. (8 letters, lowercase)