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 1 If 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 2 If 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 3 Let be loxodromics. Let be a disjoint skewer set for . Assume that no words of length generate a free semigroup. Then
- either ,
- or .
Lemma 4 Let 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.