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Dense subsets of the boundary of a Coxeter system

Author: Tetsuya Hosaka
Journal: Proc. Amer. Math. Soc. 132 (2004), 3441-3448
MSC (2000): Primary 57M07, 20F65, 20F55
Published electronically: May 12, 2004
Addendum: Proc. Amer. Math. Soc. (133) 2005, 3745-3747
MathSciNet review: 2073322
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Abstract: In this paper, we investigate dense subsets of the boundary of a Coxeter system. We show that for a Coxeter system $(W,S)$, if $W^{\{s_0\}}$ is quasi-dense in $W$ and the order $o(s_0t_0)=\infty$ for some $s_0,t_0\in S$, then there exists a point $\alpha$ in the boundary $\partial\Sigma(W,S)$ of the Coxeter system $(W,S)$such that the orbit $W\alpha$ is dense in $\partial\Sigma(W,S)$. Here $W^{\{s_0\}}=\{w\in W\,\vert\,\ell(ws)<\ell(w) \text{for each} s\in S\setminus\{s_0\} \}\setminus \{1\}$. We also show that if the set $\bigcup\{W^{\{s\}}\,\vert\, s\in S \text{such that} o(st)=\infty \text{for some} t\in S\}$ is quasi-dense in $W$, then $\{w^\infty\,\vert\, w\in W \text{such that} o(w)=\infty\}$ is dense in $\partial\Sigma(W,S)$.

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Additional Information

Tetsuya Hosaka
Affiliation: Department of Mathematics, Utsunomiya University, Utsunomiya, 321-8505, Japan

Keywords: Boundaries of Coxeter groups
Received by editor(s): April 15, 2003
Received by editor(s) in revised form: August 4, 2003
Published electronically: May 12, 2004
Additional Notes: The author was partly supported by a Grant-in-Aid for Scientific Research, The Ministry of Education, Culture, Sports, Science and Technology, Japan, (No. 15740029)
Communicated by: Ronald A. Fintushel
Article copyright: © Copyright 2004 American Mathematical Society

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