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

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.

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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)

Khek Lun Harold Chao

Abstract