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Kevin Wortman

Publications and source records attributed to Kevin Wortman.

18 recordsLinked to original sources
arXiv2017

Semidualities from products of trees

Let $K$ be a global function field of characteristic $p$, and let $Γ$ be a finite-index subgroup of an arithmetic group defined with respect to $K$ and such that any torsion element of $Γ$ is a $p$-torsion element. We define semiduality groups, and we show that $Γ$ is a $\mathbb{Z}[1/p]$-semiduality group if $Γ$ acts as a lattice on a product of trees. We also give other examples of semiduality groups, including lamplighter groups, Diestel-Leader groups, and countable sums of finite groups.

math.GR↗
arXiv2013

Horospherical limit points of S-arithmetic groups

Suppose Gamma is an S-arithmetic subgroup of a connected, semisimple algebraic group G over a global field Q (of any characteristic). It is well known that Gamma acts by isometries on a certain CAT(0) metric space X_S that is a Cartesian product of Euclidean buildings and Riemannian symmetric spaces. For a point p on the visual boundary of X_S, we show there exists a horoball based at p that is disjoint from some Gamma-orbit in X_S if and only if p lies on the boundary of a certain type of flat in X_S that we call "Q-good." This generalizes a theorem of G.Avramidi and D.W.Morris that characterizes the horospherical limit points for the action of an arithmetic group on its associated symmetric space.

math.GR↗
arXiv2012

A Durable Flash Memory Search Tree

We consider the task of optimizing the B-tree data structure, used extensively in operating systems and databases, for sustainable usage on multi-level flash memory. Empirical evidence shows that this new flash memory tree, or FM Tree, extends the operational lifespan of each block of flash memory by a factor of roughly 27 to 70 times, while still supporting logarithmic-time search tree operations.

cs.DS↗
arXiv2011

Filling boundaries of coarse manifolds in semisimple and solvable arithmetic groups

We provide partial results towards a conjectural generalization of a theorem of Lubotzky-Mozes-Raghunathan for arithmetic groups (over number fields or function fields) that implies, in low dimensions, both polynomial isoperimetric inequalities and finiteness properties. As a tool in our proof, we establish polynomial isoperimetric inequalities and finiteness properties for certain solvable groups that appear as subgroups of parabolic groups in semisimple groups, thus generalizing a theorem of Bux. We also develop a precise version of reduction theory for arithmetic groups whose proof is, for the most part, independent of whether the underlying global field is a number field or a function field.

math.GR↗
arXiv2009

Quasiflats with holes in reductive groups

We give a new proof of a theorem of Kleiner-Leeb: that any quasi-isometrically embedded Euclidean space in a product of symmetric spaces and Euclidean buildings is contained in a metric neighborhood of finitely many flats, as long as the rank of the Euclidean space is not less than the rank of the target. A bound on the size of the neighborhood and on the number of flats is determined by the size of the quasi-isometry constants. Without using asymptotic cones, our proof focuses on the intrinsic geometry of symmetric spaces and Euclidean buildings by extending the proof of Eskin-Farb's quasiflat with holes theorem for symmetric spaces with no Euclidean factors.

math.GT↗
arXiv2008

On unipotent flows in H(1,1)

We study the action of the horocycle flow on the moduli space of abelian differentials in genus two. In particular, we exhibit a classification of a specific class of probability measures that are invariant and ergodic under the horocycle flow on the stratum H(1,1).

math.DS↗
arXiv2008

Connectivity Properties of Horospheres in Euclidean Buildings and Applications to Finiteness Properties of Discrete Groups

Let G(O_S) be an S-arithmetic subgroup of a connected, absolutely almost simple linear algebraic group G over a global function field K. We show that the sum of local ranks of G determines the homological finiteness properties of G(O_S) provided the K-rank of G is 1. This shows that the general upper bound for the finiteness length of G(O_S) established in an earlier paper is sharp in this case. The geometric analysis underlying our result determines the conectivity properties of horospheres in thick Euclidean buildings.

math.GR↗
arXiv2008

Quasi-isometries of rank one S-arithmetic lattices

We complete the quasi-isometric classification of irreducible lattices in semisimple Lie groups over nondiscrete locally compact fields of characteristic zero by showing that any quasi-isometry of a rank one S-arithmetic lattice in a semisimple Lie group over nondiscrete locally compact fields of characteristic zero is a finite distance in the sup-norm from a commensurator.

math.GR↗
arXiv2005

A finitely-presented solvable group with a small quasi-isometry group

We exhibit a family of infinite, finitely-presented, nilpotent-by-abelian groups. Each member of this family is a solvable S-arithmetic group that is related to Baumslag-Solitar groups, and everyone of these groups has a quasi-isometry group that is virtually a product of a solvable real Lie group and a solvable p-adic Lie group. In addition, we propose a candidate for a polycyclic group whose quasi-isometry group is a solvable real Lie group, and we introduce a candidate for a quasi-isometrically rigid solvable group that is not finitely presented. We also record some conjectures on the large-scale geometry of lamplighter groups.

math.GR↗