Cubical accessibility and bounds on curves on surfaces
Joint with Nir Lazarovich.
Cutting a genus surface along essentially independent simple closed curves implies . Cutting a close 3-manifold along essential independent 2-spheres implies a bound (Kneser). Cutting it along nonparallel incompressible surfaces implies a bound as well (Haken).
What about intersecting curves on a surface ? No bound, unless one puts extra assumptions. Cutting along disjoint curves gives rise to an action of fundamental group on a simplicial tree. For intersecting curves, Sageev introduced a natural 2-complex on which the fundamental group acts. Realize curves by hyperbolic geodesics, assume that in the lifted configuration of lines in hyperbolic plane, lines intersect only 2 by 2. Then they split the plane into polygons. The dual graph is a quadrangulation of the plane, whence a square complex (one square per intersection). More generally, to a pattern of closed curves which, in the universal cover, intersect at most by (let us call this a -pattern), an action on a -complex is associated.
Question. Is there an apriori bound on the number of curves in a -pattern ?
Answer is positive. Fix a triangulation of the surface. In each triangle, one sees a bunch of parallel arcs near each corner. The number of triangles gives a bound.
Dunwoody generalized this idea as follows. Let be a finitely presented group, let be a presentation complex and a 1-pattern, i.e. a collection of tracks, i.e. connected sets of arcs in triangles which continue across edges in all avalaible directions. Then the the number of nonparallel tracks in is bounded . Associated to is an action of on a -cube complex.
A -pattern is a set of tracks such that the associated cube complex has dimension .
Theorem 1 Let be a finitely presented group, a presentation complex, a -pattern on . Then the number of nonparallel tracks in is bounded by .
Conversely, to an action of on a cube complex , one can associate a -pattern on a presentation complex . Pick a vertex of for each vertex of . Extend to 1-skeleton by geodesic rays, and to higher skeleta by geodesic simplices. Pull-back the pattern of mediating hyperplanes to . Get a -pattern, with .
This is not quite the reverse of Sageev’s construction. When starting from a action on a CCC, passing to a -pattern and taking the associated CCC, the number of orbits of cubes may increase dramatically.
Dunwoody gave an example of a finitely generated group acting on a simplicial tree with finite edge stabilizers and arbitrarily large number of orbits of edges. He showed that this cannot happen for finitely presented groups. Bestvina-Feighn gave an example of an action of on trees with edge stabilizers and arbitrarily large number of orbits of edges. They showed that this cannot happen is edge stabilizers are small, meaning that they do not contain .
Say that an action of on a tree is -acylindrical if stabilizers of edges have trivial intersection provided edges sit at distance . Under this assumption, Sela obtained a bound on the number of orbits of edges. We have a generalization of this.
Theorem 2 Let be a finitely presented group acting on a -dimensional cube complex. Assume the action is -acylindrical. Then the number of orbits of hyperplanes is bounded by .