Virtually Fibering Right-Angled Coxeter Groups

Abstract

We show that certain right-angled Coxeter groups have finite index subgroups that quotient to Z with finitely generated kernels. The proof uses Bestvina-Brady Morse theory facilitated by combinatorial arguments. We describe a variety of examples where the plan succeeds or fails. Among the successful examples are the right-angled reflection groups in H 4 with fundamental domain the 120-cell or the 24-cell.

Infinitesimal Rigidity for Cubulated Manifolds

“…A state gives a way to define a map from M to S 1 . This notion was introduced in [11]. We work on C instead of M , disregarding the fact that C is only a deformation retract of M when P has ideal vertices.…”

“…These manifolds are naturally homotopicallyequivalent to a cube complex. In the past two years, perfect circle-valued Morse functions on such manifolds were discovered and studied in [10] and [3], following the work of Jankiewicz -Norin -Wise [11] based on Bestvina -Brady theory [5]. The cube complex structure lifts to the cyclic covering, giving a nice combinatorial description of such a infinite-volume manifold in terms of a periodic cube complex.…”

Virtual Algebraic Fibrations of Kähler Groups

“…(By Stallings Theorem [Sta62], this is actually equivalent to the fact that the underlying 3-manifold is virtually a surface bundle over S 1 .) This result has triggered recent interest in the study of (virtual) algebraic fibration for various classes of groups, and relevant results have appeared, including during the preparations of this manuscript, see [FGK17,JNW17].…”

“…Or, stated otherwise, whether is an extension of by a finitely generated subgroup. This condition has been conveniently referred to in [32] by saying that algebraically fibers and here we will adhere to that terminology. Again, thanks to the recent results of Agol, Wise and collaborators (see [4] for accurate statements and references) the emerging picture is that “most” freely indecomposable -manifold groups (e.g., hyperbolic groups) virtually algebraically fiber.…”

“…Again, thanks to the recent results of Agol, Wise and collaborators (see [4] for accurate statements and references) the emerging picture is that “most” freely indecomposable -manifold groups (e.g., hyperbolic groups) virtually algebraically fiber. This result has triggered recent interest in the study of algebraic fibration for various classes of groups, and relevant results have appeared, including during the preparations of this manuscript, see [24, 32, 37].…”