Folding cube complexes
With Benjamin Beeker.
1. Classical Stallings folding
free group, finitely generated subgroup. It is free, what it its rank ? is the fundamental group of a bouquet, with a string of labels along each loop, defining a map to the standard bouquet of . If two loops start with the same letter, identify them; then continue.
More generally, let be a finitely generated group acting on 2simplicial trees and , and with a -equivariant map . Action on is supposedly easy, want to understand action on , by folding -equivariantly.
Dunwoody’s resolution, as described by Beeker, belongs to this scheme.
Let be a cube complex. Let be the space of half-spaces, equipped with partial order relation (inclusion) and involution (complementary half-space). These data (pocset structure) fully characterize . 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 act on corresponding to 90° rotation along the sides of a square.
Theorem 1 Let be finitely presented group acting on CCC . Assume that all hyperplane stabilizers are finitely generated. Let be the resolution. There is a finite sequence of foldings such that the last one -embeds into .
3. Application to quasi-convex subgroups
Theorem 2 Let be a surface or free group acting on CCC . If hyperplane stabilizers are finitely generated, then is undistorted in .
Indeed, the action on the resolution is cocompact. This is preserved by successive foldings.
Example. Let act on standardly on the factor and by flipping . Then the presentation complex
Theorem 3 Let be a cubulated hyperbolic group. Let be a finitely generated subgroup of . Then is quasiconvex iff stabilizers of hyperplanes and of transverse intersections of transverse hyperplanes are finitely generated.