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)}.

Advertisements

About metric2011

metric2011 is a program of Centre Emile Borel, an activity of Institut Henri Poincaré, 11 rue Pierre et Marie Curie, 75005 Paris, France. See http://www.math.ens.fr/metric2011/
This entry was posted in Workshop lecture and tagged . Bookmark the permalink.

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s