Extensions of automorphisms of self-similar groups

4.1 General results

It is a natural question to inquire which endomorphisms of an automaton group admit a continuous extension to ( G ¯ , d ) . General topology yields the following result.

Note that, if there is a continuous extension of the function θ : G → G to its closure G ¯ , it is unique. If an automorphism admits a continuous extension, we show that the latter is also an automorphism.

We can relate uniform continuity to stabilizers in the case of automorphisms.

Recall that a subgroup 𝐻 of a group 𝐺 is characteristic if φ ⁢ ( H ) ≤ H for all φ ∈ Aut ⁡ ( G ) .

It is natural to ask whether or not an automorphism of a self-similar group is itself a restriction of a continuous automorphism of Aut ⁡ ( T A ) . The following remark is a necessary condition tying the existence of extensions to Aut ⁡ ( G ¯ ) to extensions to Aut ⁡ ( Aut ⁡ ( T A ) ) . In Subsections 4.3 and 4.4, we see examples where extensions to Aut ⁡ ( Aut ⁡ ( T A ) ) may or may not exist.

4.2 Aleshin’s automaton and automorphisms

The following example shows that condition (ii) of Proposition 4.3 does not hold for every automaton group. Let 𝐺 be the automaton group generated by Aleshin’s automaton

We claim that Stab 2 ⁡ ( G ) is not characteristic. Indeed, Vorobets and Vorobets [15] proved that 𝐺 is the free group on { λ ^ p , λ ^ q , λ ^ r } ; hence λ ^ p ↦ λ ^ r ⁢ λ ^ p , λ ^ q ↦ λ ^ q , λ ^ r ↦ λ ^ r defines an elementary Nielsen automorphism 𝜃 of 𝐺. The action of the generators of 𝐺 at depth 2 is described by the diagram

It follows easily that λ ^ p ⁢ λ ^ q ∈ Stab 2 ⁡ ( G ) but ( λ ^ p ⁢ λ ^ q ) ⁢ θ = λ ^ r ⁢ λ ^ p ⁢ λ ^ q ∉ Stab 2 ⁡ ( G ) . Therefore, Stab 2 ⁡ ( G ) is not characteristic.

4.3 The adding machine and its automorphisms

We consider now the adding machine to illustrate some of the problems already introduced and some others we intend to propose. Write φ = λ ^ p in the notation of Example 2.1. Since 𝒜 generates the infinite cyclic group G ⁢ ( A ) = ⟨ φ ⟩ , there exist only two automorphisms of G ⁢ ( A ) : the identity and the automorphism 𝜃 defined by

The adding machine takes its name from the fact that it reproduces addition of integers in binary form in the following sense: if u ∈ A 2 m , then u ⁢ φ n ∈ A 2 m is the unique integer 𝑣 in binary form such that v ~ is congruent to u ~ + n modulo 2 m , where w ~ is the word 𝑤 read in reverse order. It follows that

for every m ∈ N . But then Stab m ⁡ ( G ⁢ ( A ) ) is a characteristic subgroup of G ⁢ ( A ) for every m ∈ N , and so every automorphism of G ⁢ ( A ) admits a continuous extension to G ¯ by Lemma 4.1 and Corollary 4.5.

We also claim that every automorphism of G ⁢ ( A ) can be actually extended to an automorphism of Aut ⁡ ( T 2 ) . This is trivial for the identity automorphism, so we only have to consider 𝜃. We show that 𝜃 is the restriction to G ⁢ ( A ) of the mirror image automorphism 𝜇 of Aut ⁡ ( T 2 ) .

Each φ ∈ Aut ⁡ ( T 2 ) is fully determined by the local permutations φ u ( u ∈ A 2 * ). Let σ ∈ Aut ⁡ ( A 2 * ) swap 0 and 1. We define φ ⁢ μ ∈ Aut ⁡ ( T 2 ) by

We must prove that 𝜇 is an automorphism of Aut ⁡ ( T 2 ) . We start by showing that

for all φ ∈ Aut ⁡ ( T 2 ) and u ∈ A 2 * . We use induction on | u | . The claim holds trivially for u = ε ; hence we assume that u = v ⁢ a with v ∈ A 2 * and a ∈ A 2 , and that (4.2) holds for 𝑣. We have

By the induction hypothesis, we get

where φ v ⁢ σ ⁢ σ = σ ⁢ φ v ⁢ σ since S A 2 is abelian. Therefore, (4.2) holds.

Now let φ , ψ ∈ Aut ⁡ ( T 2 ) . For every u ∈ A 2 * , (4.1) and (4.2) yield

hence ( φ ⁢ ψ ) ⁢ μ = ( φ ⁢ μ ) ⁢ ( ψ ⁢ μ ) , and so 𝜇 is a group homomorphism. Since 𝜇 is clearly bijective, it is an automorphism of Aut ⁡ ( T 2 ) .

Finally, we show that θ = μ | G ⁢ ( A ) . Writing φ = λ ^ p , it suffices to show that φ ⁢ θ = φ ⁢ μ . Let u ∈ A 2 * . We must show that

Note that (4.1) and the fact that the local functions are in S A 2 imply

hence

On the other hand,

It follows easily from the construction of the adding machine that

Now u ⁢ σ ∈ 0 ⁢ A 2 * if and only if u ∈ 1 ⁢ A 2 * if and only if u ⁢ φ - 1 ∈ 0 ⁢ A 2 * ; hence we have φ u ⁢ σ = φ u ⁢ φ - 1 , and so (4.3) follows from (4.4) and (4.5). Therefore, θ = μ | G ⁢ ( A ) , and so every automorphism of G ⁢ ( A ) is a restriction of some automorphism of Aut ⁡ ( T 2 ) .

For θ ∈ Aut ⁡ ( G ⁢ ( A ) ) , we can also compute the subgroup of fixed points

Since φ n ⁢ θ = φ - n , we have Fix ⁡ ( θ ) = { id } , hence finitely generated. We shall see in the next section that this is not always the case.

4.4 A non-uniformly continuous automorphism

In this subsection, we prove Theorem 1.1 by describing a construction to show that a direct product of automata groups is itself an automaton group. The direct product result is well known (for example, see the construction mentioned in the survey [5, immediately after Theorem 2.2] and Remark 4.8 below). However, to the best of our knowledge, the construction described in the proof below is new and potentially of independent interest, and we will use it to show an application afterwards.

Proof

We can assume that G = G ⁢ ( A ) ≤ Aut ⁡ ( T A ) and H = G ⁢ ( B ) ≤ Aut ⁡ ( T B ) are automata groups, where the automata 𝒜 and ℬ are such that the alphabets 𝐴 and 𝐵 are disjoint. We now construct two new automata A ′ and B ′ on the same alphabet C = A ∪ B . The automaton A ′ has the same states of 𝒜 and has the same transition and output function on the elements of 𝐴, while ( b , q ) ⁢ δ A ′ = q and ( b , q ) ⁢ λ A ′ = b for all states q ∈ A ′ and b ∈ B . In a similar fashion, B ′ has the same states of ℬ and has the same transition and output function on the elements of 𝐵, while ( a , q ) ⁢ δ B ′ = q and ( a , q ) ⁢ λ B ′ = a for all states q ∈ B ′ and a ∈ A . Hence the transition and output functions behave as shown in the picture below:

Let 𝒞 be the automaton given by the disjoint union of A ′ and B ′ . Observe that G ⁢ ( A ′ ) = ⟨ Q A ⟩ and G ⁢ ( B ′ ) = ⟨ Q B ⟩ , where Q A is the set of rational maps on the alphabet 𝐴 given by taking each state of 𝒜 as the initial state of the automaton. We also define Q B accordingly. By construction of A ′ and B ′ , it is clear that g ⁢ h = h ⁢ g for all g ∈ G and h ∈ H and that G ∩ H = { 1 } ; therefore, we have that G ⁢ ( C ) ≅ G × H , and so G × H is an automaton group. ∎

We will now use Proposition 4.3 and Lemma 4.7 to construct an example of a non-uniformly continuous automorphism of an automaton group. Consider the following automaton 𝒜 obtained by taking the adding machine automaton and, at each state, replacing the local permutation with its inverse (this construction is what is usually referred to as the inverse automaton):

and let 𝑝 be the rational function arising from the automaton with initial state 𝑝. By construction,

and so, since each image through 𝑝 will have a tail given by 01 ω , it follows that 𝑝 has infinite order and G ⁢ ( A ) = ⟨ p ⟩ ≅ Z . Consider the following automaton ℬ:

and let 𝑟 be the rational function coming from the automaton with initial state 𝑟. By construction,

and so, since each image through 𝑝 will have a tail given by 24 ω , 𝑟 has infinite order and G ⁢ ( B ) = ⟨ r ⟩ ≅ Z . If we build the automaton 𝒞 as in the proof of Lemma 4.7, then G ⁢ ( C ) = ⟨ p , r ⟩ ≅ Z 2 . Let θ ∈ Aut ⁡ ( G ⁢ ( C ) ) be the automorphism swapping 𝑝 with 𝑟. We argue by contradiction that 𝜃 is not uniformly continuous. By Proposition 4.3, there exists a positive integer 𝑛 such that

Observe that, for every g ∈ Aut ⁡ ( T 2 ) , we have g 2 n = id at level 𝑛. In fact, by induction, if g 2 n - 1 fixes level n - 1 pointwise, then even if it swapped some of the children of level n - 1 , we have that ( g 2 n - 1 ) 2 = g 2 n would fix all such children. Hence, in particular, p 2 n ∈ Stab n ⁡ ( G ⁢ ( C ) ) , and therefore,

However, since 𝑟 acts as the 3-cycle ( 2 3 4 ) at level 1, it follows that ( 2 3 4 ) 2 m = id , and so 3 = | ( 2 3 4 ) | divides 2 m , which is a contradiction and implies that 𝜃 is not uniformly continuous.

Another way to rephrase the previous counterexample is to consider the profinite group Z ⁢ [ 2 ] ⊕ Z ⁢ [ 3 ] seen as the profinite closure of its dense subgroup Z ⊕ Z and where the symbol Z ⁢ [ p ] denotes the group of 𝑝-adic integers. If we consider the automorphism of Z ⊕ Z that swaps the two generators of the infinite cyclic groups ℤ, then this automorphism does not extend to a continuous automorphism of Z ⁢ [ 2 ] ⊕ Z ⁢ [ 3 ] . Thus, Theorem 1.1 is proved.

5.1 Endomorphisms of the lamplighter group

We now fix k ≥ 2 and consider the lamplighter group L k = G ⁢ ( C C k ) . We identify C k with Z k , the integers modulo 𝑘 under addition. We denote by Z k * the subgroup of units of ( Z k , ⋅ ) (i.e. the integers modulo 𝑘 coprime with 𝑘). Writing α = α 1 , we get a presentation of L k of the form

Note that ξ m 0 ⁢ α n 1 ⁢ ξ m 1 ⁢ … ⁢ α n k ⁢ ξ m k has finite order if and only if

Let Fin ⁡ ( L k ) denote the subset of all elements of L k of finite order. Then Fin ⁡ ( L k ) is an abelian normal subgroup of L k , and L k / Fin ⁡ ( L k ) is infinite cyclic. Moreover, Fin ⁡ ( L k ) is a direct sum of countably many cyclic groups of order 𝑘.

For every m ∈ Z , we write β m = ξ m ⁢ α ⁢ ξ - m . It follows easily from (5.1) that each x ∈ L k can be written uniquely in the form