Volume 14, issue 2 (2014)

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$\mathrm{CAT}(0)$ spaces with boundary the join of two Cantor sets

Khek Lun Harold Chao

Algebraic & Geometric Topology 14 (2014) 1107–1122
Abstract

We will show that if a proper complete $CAT\left(0\right)$ space $X$ has a visual boundary homeomorphic to the join of two Cantor sets, and $X$ admits a geometric group action by a group containing a subgroup isomorphic to ${ℤ}^{2}$, then its Tits boundary is the spherical join of two uncountable discrete sets. If $X$ is geodesically complete, then $X$ is a product, and the group has a finite index subgroup isomorphic to a lattice in the product of two isometry groups of bounded valence bushy trees.

Keywords
CAT(0) space, CAT(0) group, Cantor set, join, spherical join, Tits boundary, visual boundary
Mathematical Subject Classification 2010
Primary: 20F65
Secondary: 20F67, 51F99
Publication
Revised: 22 April 2013
Accepted: 6 May 2013
Published: 21 March 2014
Authors
 Khek Lun Harold Chao Department of Mathematics Indiana University Bloomington, IN 47405 USA