# Complete square complexes

### Daniel T. Wise

McGill University, Montreal, Canada

## Abstract

We study groups which act cocompactly and properly discontinuously on the direct product of two trees. This class of groups turns out to be much richer than one might expect.

An interplay is developed between the immersed flats in the complex and the subgroup separability properties of its fundamental group. This link between algebra and geometry leads to the solution of several problems concerning the residual properties of automatic groups and small-cancellation groups. In particular an explicit example is given of a compact non-positively curved square complex whose fundamental group is not residually finite. A more complicated such example is given with the property that its fundamental group has no finite quotients. The universal covers of these examples are isomorphic to the direct product of two trees. Other examples include a C(4)–T(4) small-cancellation group which is not virtually torsion-free.

## Cite this article

Daniel T. Wise, Complete square complexes. Comment. Math. Helv. 82 (2007), no. 4, pp. 683–724

DOI 10.4171/CMH/107