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).

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About metric2011

metric2011 is a program of Centre Emile Borel, an activity of Institut Henri Poincaré, 11 rue Pierre et Marie Curie, 75005 Paris, France. See http://www.math.ens.fr/metric2011/
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