Atlas: Classifying Coxeter groups with non-locally connected Cat(0) boundary. by Michael L. Mihalik

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G^3 = Geometric Groups on the Gulf coast
March 4-5, 2000
University of South Alabama
Mobile, AL, USA

Organizers
Phil Bowers, Stephen Brick, Jon Corson, Igor Mineyev

Classifying Coxeter groups with non-locally connected Cat(0) boundary.
by
Michael L. Mihalik
Vanderbilt University
Coauthors: Kim Ruane (ETH), Steven Tschantz (Vanderbilt University)

A Coxeter group G has presentation of the form

P \equiv <s₁, ... s_n: s_i²=1 for alli, (s_is_j)^m_ij=1>

Here m_ij >= 2, and m_ij is defined for some pairs (i, j) with 1 <= i < j <= n.

The Coxeter graph \Gamma(P) has as vertex set, the generators of P and an edge between s_i and s_j if (s_is_j)^m_ij=1 is a relation of P.

We define an elementary graph theoretical condition on \Gamma(P) and prove that if \Gamma(P) satisfies this condition, then every Cat(0) space on which G acts geometrically has non-locally connected boundary. We also discuss our progress on proving the converse of this result. Several new results on the centralizer of a generator of P will be discussed.

Date received: February 8, 2000