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Graham Niblo works on geometric group theory. With Indira Chatterji, he explained a construction taking a group action on a wallspace to an action on a CAT(0) cube complex.^[ref]

Walls separate pairs of points; counting the separating walls gives a pseudometric. Consistent choices of sides become vertices of a cubical space. This transforms separation data into a geometric object on which the group acts, making combinatorial information available to nonpositive-curvature methods.^[ref]

Indira Chatterji studies geometric group theory. Her joint work with Graham Niblo develops a route from actions on a wallspace to actions on a CAT(0) cube complex.^[ref]

The distinction between a distance and a pseudometric matters here: different points can be separated by no walls. Counting walls still records enough separation information to support a cubical construction and transfer proper-action properties under the paper's hypotheses.^[ref]

For a wallspace $(S,\mathcal{W})$, defineThe wallspace axiom makes this finite. It is always a pseudometric: it is symmetric, vanishes on the diagonal and satisfies the triangle inequality. For the last claim, every wall separating $x$ and $z$ must separate $x$ from $y$ or $y$ from $z$, so counting walls gives the inequality.^[ref]

It is a metric exactly when the walls separate every pair of distinct points. For example, on $S=\{a,b,c\}$ with only the wall $\{\{a,b\},\{c\}\}$, the distinct points $a,b$ have distance zero. Identifying pairs at distance zero gives a genuine metric space.

The principal vertex of a dual cube complex sends indistinguishable points to the same vertex. The wall-count distance equals the combinatorial distance between their principal vertices, making this pseudometric useful for the wall-metric properness criterion.