** 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.

**2. POCSETs **

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. *

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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/