Kevin Wortman

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. Our main result is Theorem 4 below. Before stating it, we provide some background. 0.1. Arithmetic groups. Let K be a global field (number field or function field), and let S be a nonempty set of finitely many inequivalent valuations of K including one from each class of archimedean valuations. The ring OS ⊆ K will denote the corresponding ring of S-integers. For any v ∈ S, we let Kv be the completion of K with respect to v so that Kv is a locally compact field. Let G be a noncommutative, absolutely almost simple, K-isotropic K-group. Let G be the semisimple Lie group

Quasi-isometric rigidity of higher rank S -arithmetic lattices

We show that S-arithmetic lattices in semisimple Lie groups with no rank one factors are quasi-isometrically rigid.

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).

A geometric proof that SL2(ℤ[t,t−1]) is not finitely presented

We give a new proof of the theorem of Krstic-McCool from the title. Our proof has potential applications to the study of finiteness properties of other subgroups of SL 2 resulting from rings of functions on curves.

Infinite generation of non-cocompact lattices on right-angled buildings

Let Γ be a non-cocompact lattice on a locally finite regular right-angled building X. We prove that if Γ has a strict fundamental domain then Γ is not finitely generated. We use the separation properties of subcomplexes of X called tree-walls.

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.

SLn(ℤ[t]) is not FPn − 1

We prove the result from the title using the geometry of Euclidean buildings.

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-Farbs quasiflat with holes theorem for symmetric spaces with no Euclidean factors.

An infinitely generated virtual cohomology group for noncocompact arithmetic groups over function fields

Let G be a noncocompact irreducible arithmetic group over a global function field K of characteristic p, and let H be a finite-index, residually p-finite subgroup of G. We show that the cohomology of H in the dimension of its associated Euclidean building with coefficients in the field of p elements is infinite.

Semidualities from products of trees

Let