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Approximating Flats by Periodic Flats in CAT(0) Square Complexes
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- Journal:
- Canadian Journal of Mathematics / Volume 57 / Issue 2 / 01 April 2005
- Published online by Cambridge University Press:
- 20 November 2018, pp. 416-448
- Print publication:
- 01 April 2005
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- Article
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We investigate the problem of whether every immersed flat plane in a nonpositively curved square complex is the limit of periodic flat planes. Using a branched cover, we reduce the problem to the case of
$\mathcal{V}\mathcal{H}$-complexes. We solve the problem for malnormal and cyclonormal $\mathcal{V}\mathcal{H}$-complexes. We also solve the problem for complete square complexes using a different approach. We give an application towards deciding whether the elements of fundamental groups of the spaces we study have commuting powers. We note a connection between the flat approximation problem and subgroup separability.
