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Kim Ruane
Professor, Graduate Program Director

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Papers

The automorphism group of the free group of rank two is a CAT(0) group [pdf]
In this paper, we prove that the automorphism groupof the braid group on 4 strands acts faithfully and geometrically on a CAT(0) 2-complex. This is used to show that Aut(F_2) also acts faithfully and geometrically on a CAT(0) space because these two groups are isomorphic. This a joint paper with A. Piggott.and G. Walsh.

The automorphism group of a graph product with no SIL [pdf]
In this paper, we study the subgroup of automorphisms of a graph product of cyclic groups that is generated by the partial conjugations. This subgroup is itself a graph product of cyclic groups provided the defining graph has no SIL. In particular, if the vertex groups are all finite cyclic, one can conclude that this (finite index) subgroup of the automorphism group is CAT(0). This a joint paper with R. Charney, N.Stambaugh, A. Vijayan.

Normal forms for automorphisms of universal Coxeter groups and palindromic automorphisms of free groups [pdf]
In this paper, we explicitly construct Markov languages of normal forms for the groups in the title of this paper. This a joint paper with A. Piggott.

On the automorphisms of a graph product of abelian groups [pdf]
In this paper, we consider investigate the structure of the automorphism group of a graph product of abelian groups. In particular, this includes right-angled Artin and Coxeter groups. This a joint paper with A. Piggott and M. Gutierrez.

CAT(0) groups with specified boundary [pdf]
In this paper, I consider the question of whether the homeomorphism type of the visual boundary determines the space and/or the group for some first examples.

CAT(0) boundary of truncated hyperbolic space [pdf]
In this paper, I compute the CAT(0) boundary of truncated hyperbolic space.

Some geometric groups with rapid decay [pdf]
In this paper, Indira Chatterji and I show that any lattice in a rank one Lie group satisfies the Baum-Connes Conjecture.

Local connectivity of right-angled Coxeter group boundaries [pdf]
We give conditions on the defining graph of a right-angled Coxeter group that guarantee any CAT(0) boundary of the group must be locally connected.

CAT(0) HNN-extensions with non-locally connected boundary [pdf]
We show how to construct CAT(0) groups with non-locally connected boundary using HNN-extensions.

Amalgamated products with non-locally connected boundary [pdf]
CAT(0) We show how to construct CAT(0) groups with non-locally connecte boundary using amalgamated products. In particular, we show how to construct one-ended right-angled Coxeter groups where the Davis complex has non-locally connected boundary simply by giving conditions on the defining graph of the group.

Dynamics of the group action on the boundary of a CAT(0) space [pdf]
We investigate how an individual hyperbolic isometry in a CAT(0) group must act on the boundary of the CAT(0) space.

Boundaries of groups of the form GxH [pdf]
We prove that groups of the form GxH where G and H are both hyperbolic have unique CAT(0) boundary.

The angle question [pdf]
We investigate CAT(0) groups which contain an infinite order in the center. Without loss of generality, one can assume the group is of the form GxZ. We show that although G does not have to be quasi-convex in the space, there is a well-defined angle which G makes with the central element the comes from the action.