Atlas: Non-locally connected boundaries of Artin groups, Coxeter groups and parabolic extensions of $F

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G^3 = Geometric Groups on the Gulf coast
March 26-28, 1998
University of South Alabama
Mobile, AL, USA

Organizers
Stephen Brick, Jon Corson

Non-locally connected boundaries of Artin groups, Coxeter groups and parabolic extensions of F₂ by Z
by
Michael L. Mihalik
Vanderbilt University
Coauthors: Kim Ruane

Non-locally connected boundaries of Artin groups, Coxeter groups and parabolic extensions of F₂ by Z

G. Swarup has recently shown that the boundary of every one-ended word hyperbolic (negatively curved) group is locally connected. We discuss the following two theorems and some corollaries. Theorem 1. Suppose A, B and C are finitely generated groups and G=A*_C B acts geometrically on a CAT(0) space X. If the following conditions are satisfied, then \partialX is NOT locally connected: 1) [A:C] >= 2, [B:C] >= 3. 2) There exists s in G-C with sⁿ not in C for all n =/= 0 and sCs^-1 subset C. 3) Cx₀ is quasi-convex in X for a basepoint x₀.

Theorem 2. Suppose B[( \phi) || ( --> )]C is an isomorphism between finitely generated subgroups of the finitely generated group A and G is the HNN-extension <A, t:tbt^-1=\phi(b)>. If G acts geometrically on the CAT(0) space X then \partialX is not locally connected if:

(1): B is quasiconvex in X.
(2): There exists a in A such that aⁿ not in B for all n and aBa^-1 subset B.

Every extension of F₂ by Z is CAT(0). ``Parabolic" extensions of F₂ by Z have presentations of the form described in Theorem 2 (with A =~ Z \oplusZ and B =~ Z). Hence if a parabolic extension of F₂ by Z acts geometrically on a CAT(0) space X, then \partialX is not locally connected.

Corollary 1. If G is a right angled Artin group other than Zⁿ, and G acts geometrically on a CAT(0) space X, then \partialX is not locally connected.

If \Gamma is any finite connected graph then the corresponding right angled Coxeter group G_\Gamma has a presentation with generators the vertices of \Gamma and relations: v²=1 for all vertices v in \Gamma and the vertices v and w commute iff there is an edge between them. Corollary 2. If \Gamma is a finite graph, G_\Gamma the corresponding right angled Coxeter group and G_\Gamma acts geometrically on the CAT(0) space X, then \partialX is not locally connected if there are vertices v and w in \Gamma with the following two properties:

(1): v and w do not bound an edge of \Gamma.
(2): Lk(v) \cap Lk(w) (link taken in \Gamma) separates \Gamma with at least one vertex in \Gamma-(Lk(v) \cap Lk(w)) other than v and w.

Date received: March 24, 1998