Groups acting on dendrites
Joint with Nicolas Monod.
A dendrite is a continuum (compact metrizable connected spaces) which is locally connected and in which every two points are joined by a unique arc (continuous injective image of an interval).
Example. The end compactification of a tree is a dendrite.
Every dendrite can be metrized to become an -tree. Conversely, for every second countable -tree, there is a weaker topology (observer’s topology, i.e. the topology than makes point complements open) that turns it s end compactification into a dendrite.
Example. Start with a finite tripod. Glue a smaller tripod at the middle of each edge. Iterate. Get the Wasewski dendrite , homeomorphic to Berkovich’ projective line over .
2. Actions on dendrites
In view of the proximity of trees and dendrites, it is tempting to ask
Question. Can a Kazhdan group act non-elementarily on a dendrite ?
Elementary means that there is either a fixed point or an invariant arc.
2.1. Invariant measures
The order of a dendrite at a point is the number of connected components of the complement. It is at most countable. We speak of
- ends, with order 1,
- regular points, with order 2,
- branch points, with order .
The set of branched points is at most countable,
Proposition 1 For an action of on a dentrite , the following are equivalent.
- Action is elementary.
- There is a finite orbit.
- There is an invariant probability measure.
It follows that actions of amenable groups on dendrites are elementary.
Dendro-minimality means that no proper invariant subdendrites (i.e. closed connected subspaces).
On shows that a non-elementary action contains a unique closed minimal invariant subset which is contained in a unique minimal invariant subset which is a subdendrite.
Theorem 2 (Tits alternative) Either that action is elementary or contains a free group.
Proof. First reduce to minimal action. Then find elements with a north-south dynamics and play ping-pong.
Theorem 3 For every non-elementary action there is a canonical unitary representation of with a nonzero element .
It follows that higher rank lattices do not act non-elementarily on dendrites.
3. Back to Wazewski dendrites
Here is a characterization. Given a subset , is the unique dendrite such that
- Order of all brache points belong to ,
- For all , the set of order branch points of is arcwise dense.
Note that for every connected open subset of , is homeomorphic to .
Any finite subset defines a dendrite which is a finite tree. It inherits the structure of a finite labelled graph (labels are orders of endpoints).
Proposition 4 Two finite subsets of defining isomorphic labelled graphs are equivalent by a homeomorphism.
It follows that, for finite , the action of on is oligomorphic, i.e. it has finitely many orbits on -tuples for all .
Theorem 5 is simple. iff .
Indeed, the stabilizer of a point is a wreath product the homeomorphism groups of branches (components of complement of ).
3.1. Topologies on
Uniform convergence on and pointwise convergence on branch points coincide. Hence it is Polish.
Theorem 6 If is finite, is Kazhdan. Furthermore, it admits a finite Kazhdan set.
Say a group has property (OB) if in any isometric action on a metric space, orbits are bounded.
The prototype is . For locally compact groups, this can happen only if group is compact.
Theorem 7 has property (OB).
It follows that has property (FH) and property (FA). Note that, since is not locally compact, (FH) need not imply (T). The proof that is Kazhdan follows a different route.
3.2. Proof of property (T)
If is finite, is oligomorphic and simple. According to Evans and Tsankov, this implies property (T) with a finite Kazhdan set. Indeed, one shows that every unitary representation is a direct sum of irreducibles. Every irreducible representation is of the following form: given a finite subset , consider
For such representations, property (T) is proven by hand.
3.3. Proof of property (OB)
By transitivity on pairs of endpoints, . It suffices to show that has property (OB). Since is fixed, there is a partial order on . It is semi-linear and dense. The action on branched points is weakly transitive. The automorphism group of this order lifts to . According to Droste-Truss, this group has (OB), hence has (OB).