CAT(0) spaces with boundary the join of two Cantor sets

Chao, Khek Lun Harold

doi:10.2140/agt.2014.14.1107

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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

DOI: 10.2140/agt.2014.14.1107

Abstract

We will show that if a proper complete $CAT (0)$ 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

References

Publication

Received: 12 September 2012

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

Volume 14, issue 2 (2014)