** Groups acting on dendrites **

Joint with Nicolas Monod.

**1. Dendrites **

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 1For 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.

** 2.2. Minimality **

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 3For 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 4Two 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 5is 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 6If 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 7has 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).