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.