#### 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