** Uniform exponential growth for groups acting on cube complexes **

Joint with A.Kar.

**1. Uniform exponential growth **

Let be a finitely generated group. For a finite generating set , let denote the exponential rate of volume growth in .

Gromov asked wether exponential growth () implies uniform exponential growth ( finite, generating set ).

Wilson produced a counterexample.

**Question**. What about the subclass of groups acting on cube complexes ?

The usual trick is to produce a free subsemigroup generated by short words. Ping pong is good for that. For instance, it shows that two loxodromic isometries of a tree generate a free semigroup (up to changing them into their inverses) unless they stabilize a common line.

Theorem 1If are loxodromic isometries of a square complex. Then either there exist words of length that generate a free semigroup, or stabilize a common flat.

Corollary 2If a finitely generated group acts freely on a square complex, then either is virtually abelian or .

**Question**. What for higher dimensions ?

**Question**. What about torsion ?

Assume is generated by two elliptics with disjoint fixed point sets. Do they generate loxodromics ? Of what length ? Unclear.

**2. Loxodromic elements and hyperplanes **

Let act loxodromically on CCC . Say a hyperplane is *skewered* by if it is transverse to he axis of . Then some power of translates in the sense that some halfspace is mapped into itself.

Let be the set of hyperplanes skewered by . A disjoint skewer set is a subset of which is disjoint and invariant under .

Lemma 3Let be loxodromics. Let be a disjoint skewer set for . Assume that no words of length generate a free semigroup. Then

- either ,
- or .

Lemma 4Let be loxodromics. Let be a disjoint skewer set for . Assume that no words of length generate a free semigroup. Assume that . Then

- The axis of is parallel to every hyperplane in .
- .
- .

**3. The parallel complex of a loxodromic isometry **

Since every hyperplane skewered by and every hyperplan parallel to the axis of intersect, thet subset is peripheral to and contains the axis of has a product structure where is Euclidean.