Mathematical script capital V ℋ complexes, towers and subgroups of F × F

Let F be a free group. We explain the classification of finitely presented subgroups of F × F in geometric terms. The classification emerges as a special case of results concerning the structure of 2-complexes which are built out of squares and have the property that the link of each vertex has no reduced circuits whose length is odd or less than four. We obtain these results using tower arguments and elements of the theory of non-positively curved spaces. © 1999 Cambridge Philosophical Society