The Neretin groups (Bruno Duchesne)

Definition

The Neretin groups are a family of locally compact subgroups of homeomorphisms of the Cantor set.

Before defining them we will present an Archimedean analogue for them. The group \( \mathrm{PGL}_2({\mathbb R}) \) acts by homographies on the circle \( \mathbb S^1 \) (identified with the projective space \( \mathbb P^1({\mathbb R}) \)), and it is a locally compact subgroup of \( \mathrm{Homeo}(\mathbb S^1) \). This is a Lie group and it is reasonably well-understood. A more exotic group is obtained by taking those homeomorphism which are only piecewise homographies; however this group is not closed in \( \mathrm{Homeo}(\mathbb S^1) \), and there is no natural way to turn it into a locally compact group. (It has, however, many interesting discrete subgroups such as the Thompson groups and the « Frankenstein groups » defined by Monod).

In the totally discontinuous world, the group \( \mathrm{PGL}_2({\mathbb Q}_p) \) acts on \( \mathbb P^1({\mathbb Q}_p) \) (topologically a Cantor set) by homographies, and this action can be identified with the action on the boundary \( \partial T_p \) at infinity of its Bruhat–Tits tree \( T_p \) (a \( p+1 \)-regular tree). It will be more convenient to consider the entire group \( \mathrm{Isom}(T_p) \) of isometries of this tree, whose action also extends to \( \partial T_p \). The Neretin group \( N_p \) will be the subgroup of homeomorphisms of \( \partial T_p \) which are piecewise restictions of isometries. The totally discontinuous nature of \( \partial T_p \) allows a nice description of this.

Definition: Let \( F_1, F_2 \) be two (possibly empty) finite subtrees of \( T_p \) such that \( T_p \setminus F_i \) have the same number of components. If \( \phi \) is an isometry between \( T_p \setminus F_1 \) and \( T_p \setminus F_2 \) it induces an homeomorphism \( \phi_* \) of \( \partial T_p \). Then Neretin group \( N_p \) is the subgroup of \( \mathrm{Homeo}(\partial T_p) \) of all elements of this form.

Note that this subset is actually a subgroup because it is always possible to enlarge the finiet tree \( F_1 \) used in the definition, and thus to compose such elements inside the tree.

Simplicity

This is due to C. Kapoudjian. See https://arxiv.org/abs/1502.00991 for a nice account.

Topology of the Neretin groups

The Neretin group is not closed in the group \( \mathrm{Homeo}(\partial T_p) \): it is not hard to construct sequences of finite subtrees, and associated elements of \( N_p \) which converge to an homeomorphism which does not belong to \( N_p \). The size of the subtrees must go to infinity for this to happen, and it turns out that it is possible to make the Neretin group into a locally compact group, essentially by stating that such sequences must diverge.

Formally, this is done using the following fact:

If a topological group \( H \) is a commensurated subgroup in an abstract group \( G \) then there is a unique group topology on \( G \) such that \( H \) is an open subgroup.

Recall that a subgroup \( H \le G \) is said to be commensurated if for all \( g \in G \) the intersection \( H \cap gHg^{-1} \) has finite index in \( H \). This is the case for any compact-open subgroup of \( \mathrm{Isom}(T_p) \) in \( N_p \): indeed, if \( K \) is such a subgroup and \( g \in N_p \), letting \( F \) be a finite subtree outside of which \( g \) acts by homeomorphisms, and \( K_F \) the subgroup of \( K \) fixing \( F \) pointwise we have that \( K_F \) is compact open, hence of finite index in \( K \). On the other hand \( g K_F g^{-1} \) is still contained in \( \mathrm{Isom}(T_p) \), and as it follows from the previous sentence that it is compact-open its interection with \( K \) must have finite index in both, so that in particular \( K \cap gKg^{-1} \) which contains it has finite index in \( K \).

As the subgroup \( K \) is compact and open in \( N_p \) for this topology it follows that \( N_p \) is locally compact. It is second countable as the quotient \( N_p / K \) is countable.

In this topology a sequence \( g_n \in N_p \) converges if and only if there exists a pair of finite subtrees \( F \subset T_p \) such that \( g_n \) is induced from an isomorphism \( T_p \setminus F_1 \to T_p \setminus F_2 \), with a fixed map between the sets of components for \( n \) large enough and the isomorphism between components converging (in the topology of maps between rooted trees).

Absence of lattices

The goal of this section is to explain the proof of the following theorem (https://arxiv.org/abs/1008.2911).

Theorem (Bader–Caprace–Gelander–Mozes): The Neretin group has no lattices.

The proof uses the following subgroup: let \( e \) be an edge in \( T_p \) and let \( B_n \) be the ball of radius \( n \) around \( e \). Let \( O_n \) be the subgroup of elements of \( N_p \) induced by isometries of \( T_p \setminus B_n \). Then \( O_n \subset O_{n+1} \) and we can form the subgroup
\[
O = \bigcup_{n \ge 1} O_n
\]
of \( N_p \). An hyperbolic isometry of \( T_p \) cannot belong to \( O \) it is a proper subgroup of \( N_p \). Each \( O_n \) is open and compact (the stabiliser in \( isom(T_p) \) of \( e \) is a finite-index subgroup), so that:

\( O \) is an open subgroup of \( N_p \).

The main result above then follows from the following theorem.

Theorem: The group \( O \) has no lattices.

We will explain how Bader–Caprace–Gelander–Mozes rule out cocompact lattices; the general case follows the same lines but is quite a bit more involved. In what follows \( \Gamma \) is a cocompact lattice in \( O \).

First we note that for any \( n \) the action of \( O_n \) on \( \partial B_n \) gives a surjective morphism
\[
\pi_n :\: O_n \to \mathfrak S(k_n)
\]
where \( k_n = |\partial B_n| = p^n \). Let \( \Gamma_{O_n} = \Gamma \cap O_n \), a finite group, and \( \Gamma_n = \pi_n(\Gamma_{O_n}) \). As \( \Gamma \) is discrete and \( \ker(\pi_n \) form a basis of neighbourhoods of \( \mathrm{Id} \) in \( N_p \) (in fact \( \ker(\pi_n) \) is the subgroup of the stabiliser \( K \) of \( e \) in \( \mathrm{Isom}(T_p) \) fixing \( B_n \) pointwise) it follows that \( \Gamma_n \cong \Gamma_{O_n} \) for \( n \) large enough. Moreover, as it is cocompact a fundamental domain for \( \Gamma \) in \( O \) will be contained in \( O_n \) for \( n \) large enough. We fix the Haar measure on \( O \) so that \( K \) has mass 1, putting:
\[
c = \mathrm{vol}(O/\Gamma),\, c_n = \mathrm{vol}(O_n/\Gamma_n),\, a_n = |\mathrm{Isom}(B_n)|
\]
these two facts then imply that:
\[
c = c_n = \frac{\mathfrak S(k_n)}{a_n}
\]
for large enough \( n \) (as \( a_n \) is the index of \( \ker(\pi_n) \) in \( K \)).

It follows that for these \( n \) we have
\[
[\mathfrak S(k_n) : \Gamma_n] = c\cdot a_n.
\]
We note that the right-hand side has bounded prime factors as \( a_n \) is a power of \( p! \). A result from finite group theory then implies the following.

The subgroup \( \Gamma_n \subset \mathfrak S(k_n) \) contains \( \mathrm{Alt}(X_n) \) for some \( X_n \subset \partial B_n \) with \( |X_n| \ge \frac{k_n}2 + 2 \).

The subset \( X_n \) must contain at least two pairs \( (x_i, y_i) \) of leaves of \( B_n \) such that \( x_i \) and \( y_i \) have the same parent in \( B_n \). The permutation
\[
(x_1\, y_1)(x_2\, y_2)
\]
is then induced by a nontrivial automorphism of \( B_n \) which fixes pointwise \( B_{n-1} \). But for \( n \) large enough this is a contradiction to the fact that such automorphisms do not intersect \( O_n \).

The cocompact case uses similar argument, but the finite group theory involved is more complicated as in this case the equality \( [\mathfrak S(k_n) : \Gamma_n] = c\cdot a_n \) does not hold, only the asymptotic statement \( [\mathfrak S(k_n) : \Gamma_n] \sim c\cdot a_n \).

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