An automata group of intermediate growth and exponential activity

Proposition 1.

The automata group G has exponential activity and intermediate growth:

Moreover, G is a torsion group.

Proof.

The group G obviously has exponential activity because actb⁢(ℓ)=2ℓ. First observe that a2=b3=id. The key point is that 1 is a fixed point of the permutation of a and 7,8 are fixed points of the permutation of b. It follows that G is a quotient of the free product ℤ/2⁢ℤ∗ℤ/3⁢ℤ. Any minimal length representative word of some element of G has the form

with εi∈{-1,0,1}. Moreover, the positive integers pi,qi (possibly p1 or qk are zero), are at most 7, with the exception of the following four words: (a⁢b)8, its a-conjugate (b⁢a)8 and the inverses (b-1⁢a)8 and (a⁢b-1)8. Indeed, using the relation (a⁢b)16=id, we would contradict minimality.

For i from 1 to 8, denote by wi the ℤ/2⁢ℤ∗ℤ/3⁢ℤ-free reduction of the ith section of w. Set L⁢(w)=∑i=18|wi|a, where |w|a is the number of letters a appearing in w. Observe that 2⁢|w|a is the length of w up to ±1.

To check this claim, observe that each letter a in w contributes at most one letter a in one of the sections wi. Moreover, the equality

ensures that whenever the subword a⁢b⁢a⁢b-1⁢a appears in w, the subword a2=id appears in a section, reducing its length by 2. The claim follows by counting the frequency of non-overlapping such subwords. A worse-case situation is given by (a⁢b)7⁢a⁢b-1⁢a⁢b⁢(a⁢b-1)7. Note that the key point in obtaining (0.2) is that the permutation of b maps 1 to 2, and that both 1 and 2 are fixed by the permutation of a.

The claim ensures that the ball B⁢(n) of radius n in the Cayley graph of G embeds into a union of products B⁢(n1)×…×B⁢(n8), where the union is over all possible ni satisfying n1+…+n8≤78⁢n+c, of which there is polynomial choice. The precise bound on growth follows by Muchnik and Pak’s growth theorem [13].

To prove that G is torsion, we proceed by induction on |w|a, where w is a reduced word of the form (0.1). Assume that |w|a>8. Let c be a cycle of the permutation of w, of length k. For i a letter in the cycle, wk fixes i and its i⁢th section is ∏k=0k-1wσk⁢(i) of length ≤78⁢|w|a+1 by the claim. By induction, some power of wk is trivial on c. As this holds for each cycle, w is torsion. It remains to check that all w with |w|a≤8 are torsion elements, and this can be done using GAP [19].

We must also show that G does not have polynomial growth. Otherwise, it would be virtually nilpotent by Gromov’s theorem [12], hence virtually torsion-free. Therefore, it is sufficient to prove that G is infinite. This is the case because the section map g↦g1 from StabG⁢(1) to G is onto, where StabG⁢(1) is the stabiliser in G of the vertex 1 in the tree. Indeed, its image contains the generators a and b, as easily checked using the definition of a and

Remark 3.

The choice of the permutations of a and b follows a construction due to Wilson of groups of non-uniform exponential growth [18, 17]. His construction also provides groups of intermediate growth as explained in [6, Section 6]. One of them is the automata group H=〈a,b′〉<Aut⁢(T8) generated by a and

The group H only has bounded activity. It also has intermediate growth because relation (0.2) still holds. However, the best known upper bound is

see [6, Remark 6.7] which uses an argument of Erschler [7]. Perhaps counter-intuitively, augmenting the activity by replacing b′ by b permits us to obtain a better upper bound on growth, because it gives the new relation (a⁢b)16=id, bounding the values of pi,qi in (0.1).

Remark 4.

The first Grigorchuk group of intermediate growth is usually defined via the following automaton of bounded activity: a=〈id,id〉⁢(12), b=〈c,a〉, c=〈d,a〉, d=〈b,id〉. Thibault Godin pointed out to the author that replacing all the automorphisms a in this automaton by the automorphism a′=〈a′,a′〉⁢(12), which has exponential activity, yields an isomorphic copy. It is not known if the group of Proposition 1 can be generated by an automaton with bounded activity.