CAT(0) groups and Coxeter groups whose boundaries are scrambled sets

Abstract

Communicated by C.A. Weibel MSC: 20F65 20F55 57M07 a b s t r a c tIn this paper, we study CAT(0) groups and Coxeter groups whose boundaries are scrambled sets. Suppose that a group G acts geometrically (i.e. properly and cocompactly by isometries) on a proper CAT(0) space X . (Such a group G is called a CAT(0) group.) Then the group G acts by homeomorphisms on the boundary ∂X of X and we can define a metric d ∂X on the boundary ∂X. The boundary ∂X is called a scrambled set if, for any α, β ∈ ∂X with α = β, (1) lim sup{d ∂X (gα, gβ) | g ∈ G} > 0 and (2) lim inf{d ∂X (gα, gβ) | g ∈ G} = 0. We investigate when boundaries of CAT(0) groups (and Coxeter groups) are scrambled sets.

On boundaries of coxeter groups and topological fractal structures

“…Also then (the action of G on) qX is scrambled; that is, for any two points a; b A qX with a 0 b, lim supfd qX ðga; gbÞ j g A Gg > 0 and lim inffd qX ðga; gbÞ j g A Gg ¼ 0 (cf. [21]). Indeed lim supfd qX ðga; gbÞ j g A Gg > 0 always holds ([21, Theorem 3.1]) and if we take F ¼ fa; bg then for any small open subset U of qX , gF H U for some g A G, hence lim inffd qX ðga; gbÞ j g A Gg ¼ 0.…”

“…We can find recent research on minimality and scrambled sets of boundaries of Coxeter groups in [20] and [21].…”

On boundaries of Coxeter groups and topological fractal structures

“…Also then (the action of G on) ∂X is scrambled; that is, for any two points α, β ∈ ∂X with α = β, lim sup{d ∂X (gα, gβ) | g ∈ G} > 0 and lim inf{d ∂X (gα, gβ) | g ∈ G} = 0 (cf. [21]). Indeed lim sup{d ∂X (gα, gβ) | g ∈ G} > 0 always holds ([21, Theorem 3.1]) and if we take F = {α, β} then for any small open subset…”