Groups acting on rooted trees

Let d ≥ 2 be an integer and letTd be thed-ary infinite rooted tree. If we consider the alphabetX ={0, . . . , d−1}, vertices inT_dare in bijection with the setX^∗ of finite words inX. The root is represented by the empty word∅and, ifvrepresents a vertex,virepresents itsi-th child, fori∈X. Forn≥0, then-th level ofTdis the set of vertices ofTdwhich are at distancenfrom the root. It is in bijection with the setXⁿof words onXof lengthn.

The group of all graph automorphisms ofT_dis denotedAut(T_d). Any automorphism ofT_dmust fix the root, as it is the only vertex with degreed, and hence must mapXⁿto itself, for everyn≥0. Any automorphismg ∈Aut(T_d)can be inductively described by the permutationτ ∈Sym(X)it induces on the vertices of the first level and its projections gi ∈Aut(T_d),i= 0, . . . , d−1, to thedsubtrees attached at the root. Symbolically it can be written as

g=τ(g0, . . . , gd−1), where, for everyv=v0. . . vn−1 ∈Xⁿ,

g(v₀. . . vn−1) =τ(v₀)g_v0(v₁. . . vn−1).

We may extend this to any vertex of the tree by defining theprojectiong_v of an element g∈Aut(Td)on a vertexv∈X^∗as the automorphism satisfying, for everyw∈X^∗,

g(vw) =g(v)g_v(w).

Some natural subgroups to look at in the context of groups acting on rooted trees are vertex stabilizers. ForG≤Aut(T_d), thestabilizer of a vertexv∈Xⁿis the subgroup

Stab_G(v) ={g∈G|g(v) =v}.

Furthermore, we consider thelevel stabilizers, which are the subgroups StabG(n) = \

v∈Xⁿ

StabG(v),

for everyn≥1. Notice that they are of finite index for anyG∈Aut(T_d).

In addition to the treeT_d, we consider itsboundary∂T_d, which is the set of infinite, non-backtracking paths from the root. This is a compact space that arises naturally from the structure ofTd. The boundary∂Tdis in bijection with the setX^Nof infinite sequences in the setX^N. The action of a group onT_dcan be naturally extended to its boundary∂T_ddue to the fact that any automorphism ofT_dpreserves prefixes. As for vertices of the tree, we also consider the stabilizers of points in the boundary. The stabilizer of a pointξ ∈X^Nis the subgroup

Stab_G(ξ) ={g∈G|g(ξ) =ξ}.

Due to the self-similar nature of the tree, there is a natural transformation of the boundary which corresponds to identifying any of the subtrees rooted at the first level with the whole tree. This is theshiftmapσon the boundary of the tree, and corresponds to delete the first digit of any point in the boundary. Namely,

σ : X^N → X^N

ξ₀ξ₁ξ₂. . . 7→ ξ₁ξ₂. . . .

Finally, let us introduce some broad families of groups acting on rooted trees. The family of spinal groups that we consider throughout this thesis is not fully contained in any of them, but most of its remarkable examples lie precisely in the intersection of these families.

LetG≤ Aut(T_d)be a group acting onT_d. We say thatGisself-similarif for every g∈Gand everyv∈X^∗, its projectionsgv belong toG. Equivalently, if for everyi∈X, its projectionsgibelong toG.

Another class of groups acting on rooted trees we would like to introduce is that of automata groups. In order to do so, we first need to define Mealy automata.

Afinite-state Mealy automatonis a tupleA = (Q,Σ, φ, ψ)such thatQis a finite set called the set of states,Σis a finite alphabet,φ:Q×Σ→Qis a map called the transition function andψ:Q×Σ→Σis another map called the output function. We call an automaton Ainvertible if, for everyq ∈ Q, the mapψq : Σ → Σ, defined asψq(x) = ψ(q, x), is a bijection ofΣ.

A very convenient way of representing automata is via a directed graph, with set of verticesQ, and the following edges: for everyq∈Qandx∈Σ, there is an edge fromqto φ(q, x), labeled byx|ψ(q, x).

Every stateq ∈Qof an invertible automatonAinduces an automorphism of the rooted treeT_d, withd=|Σ|, which we abusively denoteqas well, recursively given byq(∅) =∅ andq(xv) =ψ(q, x)φ(q, x)(v), for everyx∈Σ,v∈Σ^∗. We say that the group generated by an invertible automatonAis the groupGA ≤Aut(T_d)generated by the automorphisms ofT_dinduced by allq∈Q. Conversely, we say that a groupG≤Aut(T_d)is anautomaton groupif there exists an invertible automatonAsuch thatG=GA.

Example 2.1.1. Grigorchuk’s group is the subgroup ofAut(T2)generated by the following automorphisms:

a=τ(1,1) b= (a, c) c= (a, d) d= (1, b)

,

where τ is the nontrivial element ofSym(X). See Figure 2.1 for an illustration of the generators. Notice the relationsa² =b² = c² = d² =bcd= 1. Grigorchuk’s group is a finitely generated infinite2-group, and it is not finitely presented. It is just infinite, i.e. any nontrivial quotient is finite. It has intermediate growth, and it is amenable but not elementary amenable. Grigorchuk’s group is self-similar and an automaton group.

a b c d

Figure 2.1: The generators of Grigorchuk’s group.

Example 2.1.2. Another well-known example of group acting on the binary tree is the Basilica group. Again settingτ to be the nontrivial permutation ofAut(X), it is generated by the following automorphisms, depicted in Figure 2.2.

a= (1, b) b=τ(a,1) .

a b

Figure 2.2: The generators of the Basilica group.

The Basilica group was introduced in [45] as a self-similar automaton group, and its Schreier graphs were fully described in [18]. It has exponential growth, becauseaandb generate a free non-abelian semigroup, and it is amenable but not elementary amenable.

Example 2.1.3. A very interesting example of a group acting on a non-binary rooted tree is the Fabrykowski-Gupta group. First introduced in [28], it remains arguably the simplest nontrivial example of group acting on the ternary tree. It is defined by the following automorphisms:

a=τ(1,1,1) b= (a,1, b) ,

withτ being the3-cycle(012)inSym(X), so thatacyclically permutes the subtrees on the first level. The generators are illustrated in Figure 2.3.

a b

Figure 2.3: The generators of the Fabrykowski-Gupta group.

The Fabrykowski-Gupta group is also a self-similar automaton group, and was shown to have intermediate growth in [6].

Example 2.1.4. The Hanoi towers group (onkpegs,k≥3) is another example of group

acting on thek-ary tree. Fork= 3, it is the subgroup ofAut(T₃)generated by a= (01)(1,1, a)

b= (02)(1, b,1) c= (12)(c,1,1)

,

see Figure 2.4 for a depiction of the generators.

a b c

Figure 2.4: The generators of the Hanoi towers group on three pegs.

The Hanoi towers groups appeared in [42] as a model of the homonymous combinatorial problem. Indeed, the Hanoi towers problem (onkpegs) can be reformulated in terms of distances between vertices in the Schreier graphs, which happen to be finite approximations of the Sierpi´nski gasket. The Hanoi towers group (onkpegs) is a self-similar automaton group for eachk≥3. It is known to be amenable but not elementary amenable fork= 3.

However, fork≥4, the question of amenability is still open.