On groups with Schottky set boundary
We study relatively hyperbolic group pairs whose boundaries are Schottky sets. We characterize the groups that have boundaries where the Schottky sets have incidence graphs with 1 or 2 components.
math.GT↗
arXiv subjects
Genevieve Walsh
Publications and source records attributed to Genevieve Walsh.
Virtual algebraic fibrations of surface-by-surface groups and orbits of the mapping class group
We show that a conjecture of Putman--Wieland, which posits the nonexistence of finite orbits for higher Prym representations of the mapping class group, is equivalent to the existence of surface-by-surface and surface-by-free groups which do not virtually algebraically fiber. While the question about the existence of such groups remains open, we will show that there exist free-by-free and free-by-surface groups which do not algebraically fiber (hence fail to be virtually RFRS).
math.GT↗
On groups with $S^2$ Bowditch boundary
We prove that a relatively hyperbolic pair $(G,P)$ has Bowditch boundary a 2-sphere if and only if it is a 3-dimensional Poincare duality pair. We prove this by studying the relationship between the Bowditch and Dahmani boundaries of relatively hyperbolic groups.
math.GR↗
Incoherence of free-by-free and surface-by-free groups
Let $G$ be the semidirect product $Γ\rtimes F_2$ where $Γ$ is either the free group $F_n$, $n > 1$ or the fundamental group $S_g$ of a closed surface of genus $g > 1$. We prove that $G$ is incoherent, solving two problems posed by D. Wise. This implies an affirmative answer to a question of J. Hillman on the fundamental group of a surface bundle over a surface. Although many groups have been shown to be incoherent using virtual algebraic fibering, we also show that not every free-by-free group virtually algebraically fibers.
math.GR↗
Incoherence and fibering of many free-by-free groups
We show that free-by-free groups satisfying a homological criterion, which we call excessive homology, are incoherent. This class is large in nature, including many examples of hyperbolic and non-hyperbolic free-by-free groups. We apply this criterion to finite index subgroups of $F_2\rtimes F_n$ to show incoherence of all such groups, and to other similar classes of groups. Furthermore, we show that a large class of groups, including free-by-free, surface-by-surface, and finitely generated-by-RAAG, algebraically fiber if they have excessive homology.
math.GR↗
Right-angled Coxeter groups with non-planar boundary
We investigate the planarity of the boundaries of right-angled Coxeter groups. We show that non-planarity of the defining graph does not necessarily imply non-planarity of every boundary of the associated right-angled Coxeter group, although it does in many cases. Our techniques yield a characterization of the triangle-free defining graphs such that the associated right-angled Coxeter group has boundary a Menger curve.
math.GT↗
Boundaries of Kleinian groups
We review the theory of splittings of hyperbolic groups, as determined by the topology of the boundary. We give explicit examples of certain phenomena and then use this to describe limit sets of Kleinian groups up to homeomorphism.
math.GT↗
Great circle links in the three-sphere
We investigate great circle links in the three-sphere, the class of links where each component is a great circle. Using the geometry of their complements, we classify such links up to five components. For any two-bridge knot complement, there is a finite cover that is the complement of a link of great circles in $S^3$. We show that for many two-bridge knots, this cover contains a closed incompressible surface. Infinitely many fillings of the two-bridge knot lift to fillings of great circle link where the incompressibility of this surface is preserved. Using this, we show that infinitely many fillings of an infinite class of two-bridge knot complements are virtually Haken.
math.GT↗
Double bubbles in the 3-torus
We present a conjecture, based on computational results, on the area minimizing way to enclose and separate two arbitrary volumes in the flat cubic 3-torus. For comparable small volumes, we prove that an area minimizing double bubble in the 3-torus is the standard double bubble from R^3.
math.DG↗