## Notes of Benjamin Beeker’s Cambridge lecture 10-01-2017

Cubical accessibility and bounds on curves on surfaces

Joint with Nir Lazarovich.

1. Introduction

Cutting a genus ${g}$ surface along ${N}$ essentially independent simple closed curves implies ${N\leq 3g-3}$. Cutting a close 3-manifold ${M}$ along ${N}$ essential independent 2-spheres implies a bound ${N\leq C(M)}$ (Kneser). Cutting it along ${N}$ nonparallel incompressible surfaces implies a bound ${N\leq C'(M)}$ 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 ${CAT(0)}$ 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 ${d}$ by ${d}$ (let us call this a ${d}$-pattern), an action on a ${CAT(0)}$ ${d}$-complex is associated.

Question. Is there an apriori bound ${C(g,d)}$ on the number of curves in a ${d}$-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 ${G}$ be a finitely presented group, let ${X}$ be a presentation complex and ${P}$ 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 ${N}$ of nonparallel tracks in ${P}$ is bounded ${N\leq C(G)}$. Associated to ${P}$ is an action of ${G}$ on a ${CAT(0)}$-cube complex.

2. Result

A ${d}$-pattern is a set of tracks such that the associated ${CAT(0)}$ cube complex has dimension ${d}$.

Theorem 1 Let ${G}$ be a finitely presented group, ${X}$ a presentation complex, ${P}$ a ${d}$-pattern on ${X}$. Then the number of nonparallel tracks in ${P}$ is bounded by ${C(G,d)}$.

Conversely, to an action of ${G}$ on a ${CAT(0)}$ cube complex ${Y}$, one can associate a ${d}$-pattern on a presentation complex ${X}$. Pick a vertex of ${Y}$ for each vertex of ${\tilde X}$. Extend to 1-skeleton by geodesic rays, and to higher skeleta by geodesic simplices. Pull-back the pattern of mediating hyperplanes to ${X}$. Get a ${d}$-pattern, with ${d\leq dim(Y)}$.

This is not quite the reverse of Sageev’s construction. When starting from a action on a CCC, passing to a ${d}$-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 ${F_2}$ on trees with ${F_2}$ 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 ${F_2}$.

Say that an action of ${G}$ on a tree is ${K}$-acylindrical if stabilizers of edges have trivial intersection provided edges sit at distance ${\geq K}$. Under this assumption, Sela obtained a bound ${C(G,K)}$ on the number of orbits of edges. We have a generalization of this.

Theorem 2 Let ${G}$ be a finitely presented group acting on a ${d}$-dimensional ${CAT(0)}$ cube complex. Assume the action is ${K}$-acylindrical. Then the number of orbits of hyperplanes is bounded by ${C(G,d,K)}$.