This talk presents the preprint https://arxiv.org/abs/1801.05801, joint work with László Márton Tóth.

Groups of automorphisms of rooted trees

Let \( T \) be a \( d \)-regular rooted tree (each vertex has \( d \) children). We assume that \( T \) is embedded in the plane, equivalently the children of each vertex are totally ordered. Let \( \mathrm{Aut}_f(T) \) be the group of finitary isomorphisms of \( T \), i.e. those automorphisms that are order-preserving on the children for all but finitely many vertices.

Such an an automorphism is prescribed by an automorphism of a finite subtree, and extended by order-preserving isomorphisms between the components of its complement. We say that it is elementary if there exists a vertex such that the automorphism permutes its children, and acts as the identity everywhere else. Such automorphisms generate \( \mathrm{Aut}_f(T) \); equivalently \( \mathrm{Aut}_f(T) \) is isomorphic to an iterated wreath product of copies of the symmetric group \( \mathfrak S(d) \).

We say that an elementary automorphism with associated permutation \( \sigma \in \mathfrak S(d) \) is alternating if \( \sigma \) is alternating. The subgroup of \( \mathrm{Aut}_f(T) \) generated by such automorphisms is called the finitary alternating automorphism group and denoted by \( \mathrm{Alt}_f(T) \).

In the sequel we will always assume that \( d \ge 5 \). The countable group \( \Gamma = \mathrm{Alt}_f(T) \) then has the following property:

Any normal subgroup of \( \Gamma \) fixes a ball around the root (in other words it is the « stabiliser of a level » of \( T \)).

Invariant random subgroups

Examples

Let \( \overline\Gamma \) be the completion of \( \Gamma \) in \( \mathrm{Aut}(T) \). It is a compact group, and as such it has an invariant measure \( \nu \) on the boundary \( \partial T \), the pushforward of the normalised Haar measure of \( \overline\Gamma \), which is in fact the visual measure from the root. It follows that the stabiliser in \( \Gamma \) of a \( \nu \)-random point of \( \partial T \) is an IRS of \( \Gamma \).

More generally, if \( C \) is a closed subset of \( \partial T \) we can define an invariant random subgroup in \( \Gamma \) as follows: take a random \( g \in \Gamma \), and then take the (pointwise) stabiliser in \( \Gamma \) of \( gC \). Note that this IRS can be written as an « intersection » of those obtained in the previous paragraph (as \( \mathrm{Stab}_\Gamma(gC) = \bigcap_{x \in C} \mathrm{Stab}_\Gamma(gx) \)). This construction yields a priori lots of distinct invariant random subgroups, as there are plenty of \( \overline\Gamma \)-orbits on \( 2^{\partial T} \).

The main result is as follows.

Theorem: Let \( H \) be a nontrivial ergodic IRS of \( \Gamma \), and assume that almost surely \( H \) has no global fixed point on \( \partial T \). Then \( H \) is induced from a finite-index subgroup of \( \Gamma \).

For example, let \( d’ < d \) and \( T' \subset T \) a \( d' \)-regular subtree. Then the subgroup of \( \mathrm{Alt}_f(T) \) preserving \( T' \) must have a global fixed point (as its conjugacy class supports an IRS): it is possible to see by elementary means that it is in fact equal to the subgroup fixing \( T' \) pointwise.

A more descriptive result is as follows.

Theorem: Let \( H \) be a nontrivial ergodic IRS of \( \Gamma \). There exists a closed subset \( C \) of \( \partial T \) such that the distribution of the fixed subset of \( H \) is the same as that of a random translate of \( C \).

This is not a complete classification, as different IRSs of \( \Gamma \) might have the same distribution for their fixed subsets. However there is a universal construction as follows: let \( C \) be the subset given by the theorem, and \( T_C \) the subtree of \( T \) such that \( C = \partial T_C \). For each component \( s \) of \( T \setminus T_C \) choose an integer \( m_s \). Let \( L(C, (m_s)_s) \) be the subgroup defined as follows: it fixes \( T_C \), and the ball of radius \( m_s \) around the root of \( s \) for each \( s \). Then \( L(gC, g_*(m_s)_s) \) is an invariant random subgroup (where \( g \in \overline\Gamma \) is a Haar-random element) and we have the following statement.

There exists \( (m_s)_s \) such that \( H \) contains \( L(gC, g_*(m_s)_s) \) almost surely.