** Finite topological generation **

Joint with Marc Burger.

We are interested in cocompact lattices in products of trees. They come with two projections in , and their closures . The vertex stabilizers are automatically topologically finitely generated.

Since is a compact open subgroup in , is a cocompact lattice in , hence finite topological generation follows.

It follows that projections of are never dense. Indeed, the automorphism group of a regular rooted tree is not topologically finitely generated.

**1. Groups of coloured tree automorphisms **

Let us color edges of a -reguler tree and fix a subgroup . Define

**Question**. When is topologically finitely generated?

Assume that is transitive. Let be the stabilizer of colour 1 in . Let

acts on the -sphere, and on the set of ends of .

Observe the following necessary conditions for top fin generation.

- is perfect (),
- has no fixed points at infinity.

Theorem 1is topologically finitely generated iff

- is perfect,
- has no fixed points at infinity.

Definition 2Let be a compact group. Say is PFG (for positively finitely generated) if for some , the probability that a random -tuple of elements topologically generate is positive.

This is stronger than top fin generation. For instance the profinite completion of does not have it (Kantor-Lubotzky). We shall prove that is PFG.

**2. Proof **

Relies on an idea of Meenaxi Bhattachargee. Let denote the probability that elements generate finite group . We want to estimate this for .

Theorem 3 (Bhattachargee 1994)Let be an epimorphism of finite groups. Then

where we sum over conjugacy classes of maximal subgroups of which map onto .

Indeed, being confined in a maximal subgroup is the obstacle to generate .

Set

Corollary 4Assume that for some ,

then is PFG.

** 2.1. Set up **

is a finite group (it will be ). Let act on some finite set , split into orbits . Let be perfect groups, and

Say a normal subgroup is standard if it is a product of normal subgroups of ‘s. A subgroup is clean if it does not contain any nontrivial standard subgroup.

Our main technical result is

Proposition 5Let be a clean maximal subgroup. Then one of the following holds.

- , and are nonabelian simple groups,

- is the graph of an isomorphism .
- is the normalizer in of ; there are at most conjugacy classes of such subgroups and
- , , for all , there is a normal subgroup of the form where is an abelian simple group, and
- , , is a proper subgroup…

** 2.2. Questions **

Caprace: from general principles, it follows that the set of -tuples generating always has empty interior.