arXiv subjects

Brendan Burns Healy

Publications and source records attributed to Brendan Burns Healy.

6 recordsLinked to original sources

arXiv2022

Group boundaries for semidirect products with Z

Bestvina's notion of a Z-structure provides a general framework for group boundaries that includes Gromov boundaries of hyperbolic groups and visual boundaries of CAT(0) groups as special cases. A refinement, known as an EZ-structure has proven useful in attacks on the Novikov Conjecture and related problems. Characterizations of groups admitting a Z- or EZ-structure are longstanding open problems. In this paper, we examine semidirect products of a group G with the integers. For example, we show that, if G is torsion-free and admits a Z-structure, then so does every semidirect product of this type. We prove a similar theorem for EZ-structures, under an additional hypothesis. As applications, we show that all closed 3-manifold groups admit Z-structures, as do all strongly polycyclic groups, and all groups of polynomial growth. In those latter two cases our Z-boundaries are always spheres. This allows one to make strong conclusions about the group cohomology and end invariants of those groups. In another direction, we expand upon the notion of an EZ-structure and discuss new applications to the Novikov Conjecture.

math.GT↗

arXiv2021

Cusped spaces and quasi-isometries of relatively hyperbolic groups

A group $Γ$ with a family of subgroups $\mathbb{P}$ is relatively hyperbolic if $Γ$ admits a cusp-uniform action on a proper $δ$--hyperbolic space. We show that any two such spaces for a given group pair are quasi-isometric, provided the spaces have "constant horospherical distortion," a condition satisfied by Groves--Manning's cusped Cayley graph and by all negatively curved symmetric spaces. Consequently the Bowditch boundary admits a canonical quasisymmetric structure, which coincides with the "naturally occurring" quasisymmetric structure of the symmetric space when considering lattices in rank one symmetric spaces. We show that a group $Γ$ is a lattice in a negatively curved symmetric space $X$ if and only if a cusped space for $Γ$ is quasi-isometric to the symmetric space. We also prove an ideal triangle characterization of the $δ$--hyperbolic spaces with uniformly perfect boundary due to Meyer and Bourdon--Kleiner. An appendix concerns the equivalence of several definitions of conical limit point found in the literature.

math.GR↗

arXiv2021

Wallpaper Groups and Auxetic Metamaterials

We examine a fundamental material property called Poisson's ratio, which establishes the relationship for the relative deformation of a physical system in orthogonal directions. Architects and engineers have designed advanced systems using repeating patterns that can potentially exhibit auxetic behavior, which is the property of having a negative Poisson's ratio. Because two-dimensional cross sections of each of these patterns has an associated wallpaper group, we can look for useful correlations between this geometric information and the two-dimensional response as defined by Poisson's ratio. By analyzing the data, we find two properties of the wallpaper group that correlate with more effective Poisson's ratio required for applications. This paper also contains an introduction to wallpaper groups and orbifold notation and an appendix contains some literature references recreated to use our preferred notation.

math.MG↗

arXiv2021

Pinched Curvature in Heintze Groups of Carnot-type

Rank-one symmetric spaces carry a solvable group model which have a generalization to a larger class of Lie groups that are one-dimensional extensions of nilpotent groups. By examining some metric properties of these symmetric spaces, we motivate and prove the existence of analogous left-invariant, Riemannian metrics on Heintze groups of Carnot-type. These metrics adhere to certain natural curvature pinching properties, and we show in a special case that this pinching is optimal, appealing to a result of Belegradek and Kapovitch.

math.DG↗

arXiv2019

Acylindrical Hyperbolicity of Out($W_n$)

We prove that the group of outer automorphisms of the free Coxeter group $W_n$ is acylindrically hyperbolic in the sense of Osin. As an application, we observe that any CAT(0) space admitting a geometric action by Out($W_n$) must contain a rank-one geodesic. The theorem proceeds from expanding on a well-known relationship between Out($W_n$) and the outer automorphism group of free groups.

math.GR↗

arXiv2019

Rigidity Properties for Hyperbolic Generalizations

We make a few observations on the absence of geometric and topological rigidity for acylindrically hyperbolic and relatively hyperbolic groups. In particular, we demonstrate the lack of a well-defined limit set for acylindrical actions on hyperbolic spaces, even under the assumption of universality. We also prove a statement about relatively hyperbolic groups inspired by a remark by Groves, Manning, and Sisto about the quasi-isometry type of combinatorial cusps. Finally, we summarize these results in a table in order to assert a meta-statement about the decay of metric rigidity as the conditions on actions on hyperbolic spaces are loosened.

math.GR↗