## Notes of Bruno Duchesne’s Oxford lecture 23-03-2017

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 ${{\mathbb R}}$-tree. Conversely, for every second countable ${{\mathbb R}}$-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 ${D_\infty}$, homeomorphic to Berkovich’ projective line over ${C_p}$.

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 ${x}$ 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 ${\geq 3}$.

The set of branched points is at most countable,

Proposition 1 For an action of ${G}$ on a dentrite ${X}$, the following are equivalent.

1. Action is elementary.
2. There is a finite orbit.
3. 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 ${G}$ 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 ${\pi}$ of ${G}$ with a nonzero element ${w\in H^2_b(G,\pi)}$.

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 ${S\subset \{3,4,\ldots}$, ${D_S}$ is the unique dendrite such that

1. Order of all brache points belong to ${S}$,
2. For all ${n\in S}$, the set of order ${n}$ branch points of ${D_S}$ is arcwise dense.

Note that for every connected open subset ${O}$ of ${D_S}$, ${\bar O}$ is homeomorphic to ${D_S}$.

Any finite subset ${F\subset D_S}$ defines a dendrite ${[F]}$ 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 ${D_S}$ defining isomorphic labelled graphs are equivalent by a homeomorphism.

It follows that, for finite ${S}$, the action of ${Homeo(D_S)}$ on ${D_S}$ is oligomorphic, i.e. it has finitely many orbits on ${n}$-tuples for all ${n}$.

Theorem 5 ${Homeo(D_S)}$ is simple. ${Homeo(D_S)\simeq Homeo(D_{S'})}$ iff ${S=S'}$.

Indeed, the stabilizer of a point ${x}$ is a wreath product the homeomorphism groups of branches (components of complement of ${x}$).

3.1. Topologies on ${Homeo(D_S)}$

Uniform convergence on ${D_S}$ and pointwise convergence on branch points coincide. Hence it is Polish.

Theorem 6 If ${S}$ is finite, ${Homeo(D_S)}$ 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 ${Homeo(S^1)}$. For locally compact groups, this can happen only if group is compact.

Theorem 7 ${Homeo(D_S)}$ has property (OB).

It follows that ${Homeo(D_S)}$ has property (FH) and property (FA). Note that, since ${Homeo(D_S)}$ is not locally compact, (FH) need not imply (T). The proof that ${Homeo(D_S)}$ is Kazhdan follows a different route.

3.2. Proof of property (T)

If ${S}$ is finite, ${Homeo(D_S)}$ 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 ${F\subset D_S}$, consider

$\displaystyle \begin{array}{rcl} \pi_S = \ell^2(Homeo(D_S)/Stab(F)). \end{array}$

For such representations, property (T) is proven by hand.

3.3. Proof of property (OB)

By transitivity on pairs of endpoints, ${Homeo(D_S)\subset Stab(x)\cdot Stab(y)}$. It suffices to show that ${Stab(x)}$ has property (OB). Since ${x}$ is fixed, there is a partial order on ${D_S}$. It is semi-linear and dense. The action on branched points is weakly transitive. The automorphism group of this order lifts to ${D_S}$. According to Droste-Truss, this group has (OB), hence ${Homeo(D_S)}$ has (OB).