Notes of Nir Lazarovich’s Cambridge lecture 10-01-2017

Folding {CAT(0)} cube complexes

With Benjamin Beeker.

1. Classical Stallings folding

{F} free group, {G\subset F} finitely generated subgroup. It is free, what it its rank ? {G} is the fundamental group of a bouquet, with a string of labels along each loop, defining a map to the standard bouquet of {F}. If two loops start with the same letter, identify them; then continue.

More generally, let {G} be a finitely generated group acting on 2simplicial trees {T} and {T'}, and with a {G}-equivariant map {T'\rightarrow T}. Action on {T} is supposedly easy, want to understand action on {T'}, by folding {T'} {G}-equivariantly.

Dunwoody’s resolution, as described by Beeker, belongs to this scheme.

2. POCSETs

Let {X} be a {CAT(0)} cube complex. Let {\mathcal{H}} be the space of half-spaces, equipped with partial order relation (inclusion) and involution (complementary half-space). These data (pocset structure) fully characterize {X}. This language is more convenient for the description of quotients.

Typically, we want to be able to identify halfspaces in a quotient. For this, we need that both halfspaces agree on which halfspaces they contain. Therefore they must not be separated by another identified pair.

Example. Let {G={\mathbb Z}} act on {{\mathbb R}} corresponding to 90° rotation along the sides of a square.

Theorem 1 Let {G} be finitely presented group acting on CCC {X}. Assume that all hyperplane stabilizers are finitely generated. Let {X'} be the resolution. There is a finite sequence of foldings {X'\rightarrow X_1\rightarrow\cdots\rightarrow X_k} such that the last one {L^1}-embeds into {X}.

3. Application to quasi-convex subgroups

Theorem 2 Let {G} be a surface or free group acting on CCC {X}. If hyperplane stabilizers are finitely generated, then {G} is undistorted in {X}.

Indeed, the action on the resolution is cocompact. This is preserved by successive foldings.

Example. Let {G={\mathbb Z}^2} act on {X={\mathbb R}^2\times[0,1]} standardly on the {{\mathbb R}^2} factor and by flipping {[0,1]}. Then the presentation complex

Theorem 3 Let {G} be a cubulated hyperbolic group. Let {H} be a finitely generated subgroup of {G}. Then {H} is quasiconvex iff stabilizers of hyperplanes and of transverse intersections of transverse hyperplanes are finitely generated.

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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/
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