Cayley graphs with few automorphisms: the case of infinite groups

We characterize the finitely generated groups that admit a Cayley graph whose only automorphisms are the translations, confirming a conjecture by Watkins from 1976. The proof relies on random walk techniques. As a consequence, every finitely generated group admits a Cayley graph with countable automorphism group. We also treat the case of directed graphs.


Introduction
Given a group G and a symmetric generating set S ⊆ G \ {1}, the Cayley graph Cay(G, S) is the simple unoriented graph with vertex set G and an edge between g and h precisely when g −1 h ∈ S. By construction, the action by left-multiplication of G on itself induces an action of the group on its Cayley graph, which is free and vertex-transitive.
We are here interested in the question to decide when S can be chosen so that these are all the automorphisms of Cay(G, S), or equivalently the automorphism group of the graph acts freely and transitively on the vertex set. The main result of this paper is Theorem 1.1. -Every finitely generated group G that is not virtually abelian admits a finite degree Cayley graph whose automorphism group is not larger than G acting by left-translation.
A Cayley graph of G whose automorphism group acts freely on its vertex set is called a graphical regular representation, or GRR.
The main result in [LS21] is similar to Theorem 1.1, but with the assumption of not being virtually abelian replaced by having an element of infinite (or sufficiently large) order and being non-abelian and non-generalized dicyclic (see Section 2 for definitions). Combining both results, we obtain that the infinite finitely generated groups that do not admit a GRR are precisely the abelian and the generalized dicyclic groups.
Corollary 1.2. -For a finitely generated group G, the following are equivalent: (1) G admits a GRR, (2) G admits a finite degree GRR, (3) G does not belong to the following list: • the non-trivial abelian groups different from Z/2Z and (Z/2Z) n for n 5, • the generalized dicyclic groups, • the following 10 finite groups of cardinality at most 32 (2) : the dihedral groups of order 6, 8, 10, the alternating group A 4 , the products Q 8 ×Z/3Z (1) We refer to the introduction of [LS21] and the references therein for a more detailed exposition of the work on finite groups.
(2) with GAP IDs respectively [ The first digit in the GAP ID is the order of the group, and the second is the label of the group and Q 8 × Z/4Z (for Q 8 the quaternion group) and the four groups given by the presentations a, b, c a 2 = b 2 = c 2 = 1, abc = bca = cab , a, b a 8 = b 2 = 1, b −1 ab = a 5 , a, b, c a 3 = b 3 = c 2 = (ac) 2 = (bc) 2 = 1, ab = ba , a, b, c a 3 = b 3 = c 3 = 1, ac = ca, bc = cb, b −1 ab = ac .

Consequences
Another consequence of our result is the following fact, which was a motivation of the second-named author for [LS21] and the present work, see [ST19]. This solves a conjecture raised in [ST19].
Corollary 1.3. -Every finitely generated group admits a finite degree Cayley graph whose automorphism group is countable.
Recall the classical fact that the topology of pointwise convergence on vertices turns the automorphism group of a finite degree graph into a locally compact metrizable group: a sequence (φ n ) n of automorphisms converges to φ if and only if for every x, φ n (x) = φ(x) for all but finitely many n. For this topology the stabilizer of a vertex is a compact subgroup, and therefore either finite or uncountable. Therefore Corollary 1.3 can be equivalently phrased as Every finitely generated group admits a finite degree Cayley graph whose automorphism group is discrete (equivalently has finite stabilizers). Corollary 1.3 has interesting graph-theoretical consequences, that we now explain. Following [Ben13, Geo17, ST19], given two graphs X and Y and a positive integer R, we say that Y is R-locally X if every ball of radius R in Y appears as a ball of radius R in X. A graph X is local to global rigid (LG-rigid) if there is R > 0 such that any graph that is R-locally X is covered by X.
Corollary 1.4. -Every finitely presented group admits a finite degree Cayley graph that is LG-rigid.
The existence of a LG-rigid finite degree Cayley graph is actually equivalent to finite presentability, as a non finitely presented group cannot admit LG-rigid Cayley graphs, see [ST19].

About the proof
Theorem 1.1 is only new for groups with bounded exponents: the case of groups with elements of infinite (or arbitrarily large) order was covered in [LS21]. We have in GAP's numbering of groups of that order. For example, the last group in the list is of order 27, and is the third in GAP's list of groups of order 27. It is also isomorphic to the free Burnside group B(2, 3). therefore concentrated our efforts for finitely generated torsion groups, but the proof that we finally managed to obtain turned out to apply without much more efforts in the generality of Theorem 1.1.
The proof relies partly on the results from [LS21], and partly on new ideas involving random walks on groups. In particular we use some recent results by Tointon [Toi20] on the probability that two independent realizations of the random walk commute. Necessary background of group theory is presented in Section 2. We discuss the needed contributions from [LS21] in Section 3. The new aspects, including random walk reminders are presented in Section 4. In the small Section 5, we deduce the main theorem and its corollaries, and then in Section 6 we discuss a conjecture that would significantly simplify our proofs. Finally, in Section 7 we briefly discuss some directed variants of the GRR problem.

Group theory background
This short section sets the group theoretical notation and terminology and contains some standard group theory results that will be used later. It can be safely skipped by most readers.
We start with a definition used in the introduction.
Definition 2.1. -A generalized dicyclic group is a non-abelian group with an abelian subgroup A of index 2 and an element x not in A such that x 4 = 1 and xax −1 = a −1 for all a ∈ A.
A group is said to be of exponent n if every element satisfies g n = 1, and of bounded exponent if it is of exponent n for some integer n. It is known that finitely generated groups of exponent n are finite if n ∈ {1, 2, 3, 4, 6}, but can be infinite for large n, see [Rob95,14.2].
We shall denote the index of a subgroup H of G by |G : H|.
In the following, by group property we mean of property P that a group G can have ("G is P ") or not ("G is not P ").
Definition 2.2. -If P and Q are two group properties, we say that a group G is: • P -by-Q if it admits a normal subgroup N that is P such that the quotient group G/N is Q, • locally P if every finitely generated subgroup of G is P , • virtually P if G admits a finite index subgroup H that is P .
Observe that a locally finite group G is finitely generated if and only if it is finite. A group G is 2-nilpotent (or nilpotent of class 2) if it is an extension 1 → N → G → K → 1 of abelian groups N, K with N Z(G).
We will use later the following elementary fact: is an extension with K virtually abelian and |G : C G (N )| < ∞, then G is virtually 2-nilpotent.
Proof. -If K 1 < K is abelian with finite index, then the intersection of the preimage of K 1 in G with the centralizer C G (N ) of N in G is the intersection of two finite index subgroups of G and therefore has finite index in G. It is clearly 2-nilpotent.
We also need the following folklore variant: Lemma 2.4. -If a finitely generated group is finite-by-(virtually abelian), then it is virtually abelian.
Proof. -A finite-by-(virtually abelian) group G 0 contains a finite index finite-byabelian group G (the preimage in G 0 of the finite index abelian subgroup in the quotient). So let 1 → N → G → K → 1 with N finite and K abelian. Let g ∈ G.
Since K is abelian, the conjugacy class of g is contained in gN and is thus finite; equivalently C G (g) has finite index in G. If G 0 is finitely generated, then so is G. If S is a finite generating set of G, the center of G is equal to ∩ g ∈ S C G (g) and therefore is an abelian subgroup of finite index in G. This proves that G, and hence G 0 , is virtually abelian.
We will make use of the following lemma which is due to Dicman, see for example [Rob95, 14.5.7].
Lemma 2.5 (Dicman's Lemma). -Let G be a group and X ⊆ G be a finite subset that is invariant by conjugation by elements of G and such that every g ∈ X is of finite order. Then the normal subgroup X G is finite.

Constructing Cayley graphs with few automorphisms
We follow the same general strategy for constructing Cayley graphs with few automorphisms as the one initiated in [ST19] and later developed in [LS21]. There are two independent steps.
The first step consists in finding a finite symmetric generating set S 1 of G in which the only knowledge of the colour {g −1 h, h −1 g} ⊆ S 1 of every edge {g, h} allows us to reconstruct the orientation of every edge. To say it in formulas, following [LS21] let us say that the pair (G, S 1 ) is orientation-rigid if the only permutations φ of G such that φ(gs) ∈ {φ(g)s, φ(g)s −1 } for every g ∈ G and s ∈ S 1 are the left-translations by elements of G.
The second step is, given a finite symmetric generating set S 1 ⊆ G, to find another finite symmetric generating set S 1 ⊆ S 2 such that every automorphism φ of Cay(G, S 2 ) induces a colour-preserving automorphism of Cay(G, S 1 ), that is satisfies φ(gs) ∈ {φ(g)s, φ(g)s −1 } for every g ∈ G and s ∈ S 1 . Clearly, if we are able to perform both steps and we apply the second step for S 1 that is orientation-rigid, then we obtain a Cayley graph Cay(G, S 2 ) in which the only automorphisms are the translations.
For the first step, there is nothing new to do, as we know exactly which groups admit an orientation-rigid pair (G, S 1 ) with S 1 generating. The following is a portion of [LS21, Theorem 7].
Theorem 3.1. -Let G be a finitely generated group that is neither abelian [with an element of order greater than 2] nor generalized dicyclic, and S 0 ⊆ G \ {1} be a finite symmetric generating set. Then So all the new work lies in the second step. As in [LS21,ST19], the main tool to recognize the colour of the edges is by counting triangles. If S is a finite symmetric subset of a group G, we denote by N 3 (s, S) the number of triangles in the Cayley graph Cay(G, S) containing both vertices 1 and s: The relevance of this is the following easy observation that automorphisms preserve the number of triangles of a given edge: if φ is an automorphism of Cay(G, s), then for every g, h ∈ G, Clearly, N 3 (s, S) = N 3 (s −1 , S), so counting triangles can never do more than recover the colour of edges. But if s ∈ S is such that the only elements t of S such that N 3 (s, S) = N 3 (t, S) are s and s −1 , then the automorphism group of Cay(G, S) preserves the colour of every edge corresponding to s. Our main new technical result is the following.
Proposition 3.2. -Let G be a finitely generated group that is not virtually abelian and S ⊂ G\{1} be a finite generating set. Then there exists a finite symmetric generating set S ⊆ S ⊂ G \ {1} of size bounded by 2|S|(|S| + 14) such that for all Moreover, S \ S does not have elements of order 2.
The next section is devoted to the proof of the above Proposition.

Squares and random walks
The aim of this section is to prove Proposition 3.2. We start with some discussions on the square map sq : G → G defined by sq(g) = g 2 , which will play an important role in our work. The main result of this section is Proposition 4.4, that will be proved using random walks.

On the squares in finitely generated groups
In our previous work, [LS21], we restricted our attention on groups G with an element of "big" (possibly infinite) order. In other words, we asked G to have a "big" cyclic subgroup C. The main advantage of this hypothesis is the fact that in cyclic groups sq −1 (g) consist of at most 2 elements and therefore, for any finite subset F of G the set sq −1 (F ) ∩ C is finite of size at most 2|F |. In order to generalize results from [LS21] to arbitrary infinite finitely generated groups, we first need to establish some facts on the map sq and on its fibers. We begin our analysis of the map sq by showing that infinite finitely generated groups contains infinitely many squares.
As a first consequence of Dicman's Lemma 2.5, we obtain the following Lemma 4.1. -Let n be a fixed integer in {1, 2, 3, 4, 6}. If G is an infinite finitely generated group, then {g 2 | g ∈ G, g n = 1} is infinite.
Proof. -By contradiction, suppose that the set X = {g 2 | g ∈ G, g n = 1} is finite. This set is invariant by conjugation and, since it is finite, contains only elements of finite order. Hence by Dicman's Lemma, it generates a finite normal subgroup N . But then G/N is an infinite finitely generated group in which every element satisfies g n = 1 or g 2 = 1, so is of exponent n = n if n is even and n = 2n is n is odd. In both cases, n ∈ {1, 2, 3, 4, 6}, so the free Burnside group B(m, n ) being finite (see for example [Rob95,14.2]), the group G/N is finite, which is the desired contradiction. The following elementary lemma will be useful. Proof. -We have We now state our main new contribution, which can be seen as a much stronger form of Corollary 4.2, for finitely generated groups that are not virtually abelian. The next result roughly asserts that, in such a group, not even the half of the elements can have finitely many squares.
Proposition 4.4. -Let G be a finitely generated group that is not virtually abelian. Then for every s ∈ G and F ⊂ G finite there are infinitely many g ∈ The fact that G is not virtually abelian is essential in the proposition. For example, the result is not true for the infinite dihedral group G = Z/2Z * Z/2Z. Indeed, if s is one of the generators of the free factors, then G = sq −1 (1) ∪ s sq −1 (1). Another example is provided by a generalized dicyclic group G: if x ∈ G is an element of order 4 such that the conjugation by x induces the inverse on an index 2 abelian subgroup A, then G = sq −1 (x 2 ) ∪ x sq −1 (x 2 ). This example shows that F does not necessarily contain the identity.
The proof of this proposition will rely on the following lemma, which strengthens Neumann's lemma [Neu54, Lemma 4.1].
Lemma 4.5. -Let G be a finitely generated group, s ∈ G and H 1 , . . . , H m be subgroups of G and a 1 , . . . , a m ∈ G. Assume that 0.19, then either G is virtually abelian, or the conjugacy class of s 2 is finite of cardinality less than The proof of the Proposition 4.4 will only use the lemma in the case when the H i all have infinite indices, i.e. when α = 0, but we find this quantitative form amusing.
We will prove the above lemma in the next subsection. Let us first explain how the proposition follows.
Proof of Proposition 4.4. -Let G be a finitely generated group. Suppose that there exists s ∈ G and F ⊂ G finite such that there are only finitely many g ∈ We will show that such a group is virtually abelian. We will do that in several steps. First we will show that C G (s) is locally finite. Then by multiple reductions we will prove that G is virtually 2-nilpotent and finally that G is in fact virtually abelian.
Suppose that C G (s) is not locally finite. By Corollary 4.2, C G (s) has infinitely many squares. This implies that C G (s)\(sq −1 (F )∪s sq −1 (F )) is infinite for any finite subset F ⊂ G, as otherwise this would imply that all but finitely many g ∈ C G (s) satisfy g 2 ∈ F or g 2 = s 2 (s −1 g) 2 ∈ s 2 F , i.e. C G (s) has finitely many squares.

4.1.2.
We know that C G (s) is locally finite. By assumption, there exist finite subsets E, F ⊂ G such that We will now prove that G is virtually 2-nilpotent.

Reduction to F = {1}
Let F 1 ⊆ F be the subset of elements with finite conjugacy class (the intersection of F with the FC-center of G).
The set of g ∈ G such that g(F \ F 1 )g −1 ∩ F = ∅ is a finite union of cosets of the groups C G (f ) for f ∈ F \ F 1 . So it is a finite union of cosets of infinite-index subgroups. By Neumann's lemma [Neu54,Lemma 4.1], this finite union is a strict subset of G, and there is g 0 ∈ G such that g 0 (F \F 1 )g −1 0 ∩F = ∅. For such a g 0 and for This implies that, on g 0 sq −1 (F \F 1 )g −1 0 \C G (s)E, both maps g → g 2 and g → (s −1 g) 2 take finitely many values, so by Lemma 4.3 we obtain In particular, we deduce from (4.2) that Denote by N the subgroup generated by the finite set F G 1 := ∪ g ∈ G gF 1 g −1 . Then N is normal, and its centralizer in G, which is the intersection of the centralizers of f for f ∈ F G 1 , is a finite intersection of finite index subgroups, so has finite index. Let G = G /N , and s , A , B , H be the images of s, A, B, C G (s) in G respectively. In the quotient, the previous equation becomes By Lemma 2.3, either G is virtually 2-nilpotent and we are done, or G is not virtually abelian. We can therefore suppose that G is not virtually abelian and hence infinite.

Reduction to s 2 = 1
Observe that H , the image of the locally finite group C G (s) in the quotient G/N , is locally finite, so it cannot have finite index since G is finitely generated and infinite and so are its finite index subgroups. We deduce by (1) in Lemma 4.5 that s 2 has finite conjugacy class. Moreover, C G (s) being locally finite, s 2 has finite order. By Dicman's Lemma, the normal subgroup M generated by s 2 is finite. Let G := G /M . Then G = sq −1 (1) ∪ s sq −1 (1) ∪ A H B with s 2 = 1, H an infinite index subgroup and A and B finite subsets. 4.1.5. The group G is virtually 2-nilpotent By a direct application of (2) in Lemma 4.5, we obtain that G is virtually abelian. This implies that G is finite-by-(virtually abelian) and, since it is finitely generated, virtually abelian (see Lemma 2.4). This is the desired contradiction.

The group G is virtually abelian
We already know that G is finitely generated and virtually 2-nilpotent. Let H be a finite index subgroup of G that is 2-nilpotent.
By definition, the derived subgroup ghg −1 h −1 | g, h ∈ H of H is contained in the center of H. Since H is nilpotent, we also know that the subset of torsion elements is a subgroup of H (see for example [Rob95,5.2.7]) and that all subgroups of H, and also of G, are finitely generated (see for example [Rob95,5.4.6]). In particular, since C G (s) is locally finite, it is finite. By (4.2), this implies that there exists a finite F ⊂ G such that G = sq −1 (F ) ∪ s sq −1 (F ). We claim that there exists n ∈ N such that sg n s −1 = g −n for every g in G. Indeed, let n := lcm{k | 1 k 3|F |}. Then if g has order at most 3|F | we have g n = 1 and the desired identity holds. On the other hand, let g be of order at least 3|F | + 1. For such a g, there exists at most 2|F | integers 1 k 3|F | + 1 such that g 2k is in F . Therefore, there is at least |F | + 1 integers 1 k 3|F | + 1 with (sg k ) 2 ∈ F . By the pigeonhole principle, we have 1 k = l 3|F | + 1, which is less than the order of g, such that (sg k ) 2 = (sg l ) 2 and hence sg k−l s −1 = g −(k−l) and the desired identity holds. For every g, h ∈ G, we have When g, h ∈ H, a := [g n , h n ] belongs to the center of H, so (4.4) shows that a n = 1. The subgroup K < Z(H) of elements of the center that are of finite order is finite. In the quotient H/K, the subgroup H generated by {g n | g ∈ H/K} is abelian. Moreover, the quotient (H/K)/H is a finitely generated nilpotent group of exponent n, so is finite (see for example [Rob95,5.2.18]). This implies that H, and therefore also G, is virtually abelian.

Random walks and proof of Lemma 4.5
The proof of Lemma 4.5 will use random walk techniques, and in particular the recent result of Tointon [Toi20] generalizing to infinite groups a classical result by P. Neumann [Neu89] roughly saying: a finite group in which the probability that two randomly chosen elements commute is large is almost abelian. We fix a symmetric probability measure µ on G whose support is finite, generates G and contains the identity. In particular, in this subsection G will always be a finitely generated group. Let g n and g n be two independent realizations of the random walk on G given by µ, that is two independent random variables with distribution µ * n , the n th convolution power of µ. We will use two facts. The first is very easy (3) and asserts that if H is a subgroup of G and a ∈ G, then  In particular, if H has infinite index, lim n P(g n ∈ aH) = 0. Actually, more is known: the above convergence is uniform in a and H. In the vocabulary of [Toi20], µ * n measures indices uniformly, see [Toi20, Theorem 1.11]. To illustrate the power of random walks on groups, observe that (4.5) allows to give a transparent proof (for finitely generated groups) of Neumann's lemma: if G = a 1 H 1 ∪ . . . a m H m is a finite union of cosets of subgroups, then we have The second fact we will use, [Toi20, Theorem 1.9], asserts that, whenever G is not virtually abelian (4.6) lim n P (g n and g n commute) = 0.
We start with an easy consequence of (4.6), that we will use in the proof. It is natural to expect that the result holds with replaced by an arbitrary positive number, see Section 6.
So, if c = lim inf n c n , we obtain lim inf n P g 2 n = g n 2 = (g n g n ) 2 = 1 c 2 + c − 1, which is positive if and only if c > √ 5−1 2 . To conclude using (4.6), it remains to observe that g 2 n = g n 2 = (g n g n ) 2 = 1 implies that g n and g n commute.
We now proceed to prove Lemma 4.5.
If c := lim inf n P(g 2 n = 1) > Observe that 3− √ 5−4α 2 > 0 by our assumption on α. Now, if g −1 n s 2 g n = s −2 and g n −1 s 2 g n = s −2 , then g n g n and s 2 commute, or equivalently g n g n ∈ C G (s 2 ). Therefore, since g n g n is distributed as g 2n , we obtain lim sup By (4.5), the left-hand side is equal to 1 |G:C G (s 2 )| and the claim is proven. Proof of Lemma 4.5 (2). -Let G, s, m, H i , a i satisfying (4.1), with s 2 = 1 and α < (1−α) 3 24 . Our main goal will be to prove that two randomly chosen elements of G (for wellchosen probability measures on G that are not exactly random walks but mixtures of random walks) commute with non-vanishing probability. By [Toi20], we will deduce that G is virtually abelian. We proceed by contradiction, and assume that G is not virtually abelian.
The element s having order 2, we can as well assume that the probability measure µ is s-left-invariant, that is it satisfies µ(sg) = µ(g) for every g ∈ G. By (4.1) and (4.5), we know that lim inf n P g n ∈ sq −1 (1) ∪ s sq −1 (1) 1 − α.
On the other hand, g n g n being distributed as g 2n , we have lim inf n P (g n g n ) 2 = 1 or (sg n g n ) 2 = 1 1 − α.
Let now g (1) n , g (2) n and g (3) n be three independent copies of the random walk on G given by µ. Let A n be the event A n = ∀ 1 i = j 3, g (i) n 2 = 1 and sg (i) n g (j) n 2 = 1 .

ANNALES HENRI LEBESGUE
It follows from the preceding discussion that the difference of {∀ 1 i 3, (g (i) n ) 2 = 1} and A n has probability 3α + o(1), so This is strictly positive by assumption. But on A n , for every 1 i, j 3, we have n and therefore n . To say it differently, sg (1) n commutes with g (2) n g (3) n . We deduce P sg (1) n , g (2) n g (3) Using that µ is s-invariant, sg (1) n is distributed as g (1) n and P g (1) n , g (2) n g (3) We can rewrite this as Denote by ν n the probability 1 2 (µ * n + µ * 2n ). We clearly have ν n ⊗ ν n 1 4 (µ * n ⊗ µ * 2n ), so lim inf On the other hand, it follows from [Toi20, Theorem 1.11] that the sequences of measures µ * n and µ * 2n (and therefore also ν n ) measure index uniformly, hence the preceding is a contradiction with [Toi20, Theorem 1.9]. So our starting assumption that G is not virtually abelian is absurd. This concludes the proof of the Lemma 4.5.

Counting triangles
We now state and prove a lemma on the augmentation of the number of triangles (in Cayley graphs of G) containing some s 0 ∈ G. It complements results of [ST19, Lemma 9.2] and [LS21, Lemma 31] which where valid for groups with elements of infinite (respectively very large) order. The conclusion of Lemma 4.7 is also cleaner than in [LS21,ST19].
Lemma 4.7. -Let G be a finitely generated group that is not virtually abelian, and let S ⊂ G \ {1} be a finite symmetric generating set.
Proof. -Let g be an element of G and ∆ g := {g, g −1 , s −1 0 g, g −1 s 0 }. We will show that there exists some g in G such that S = S g := S ∪ ∆ g works. Observe that for all g, the set S g satisfies Condition 1 of the lemma, and that it satisfies 2 if and only if g 2 = 1 and (s −1 0 g) 2 = 1, or equivalently g / ∈ sq −1 (1) ∪ s 0 sq −1 (1). We first restrict our attention to elements g such that the following two conditions hold |g| S 3 (4.7) where |g| S is the word length of g relative to the generating set S.
Since S is finite, the number of g ∈ G such that one of the conditions (4.7)-(4.8) do not hold is finite. Also, for a g satisfying Conditions (4.7) and (4.8) the intersection ∆ g ∩ S is empty and Condition (3) is automatically satisfied. Moreover, in the Cayley graph of G relative to S g , a triangle with a side labelled by s ∈ ∆ g has at least another side labelled by an element of ∆ g , otherwise s would have S-length at most 2. This implies that any edge labelled by s ∈ ∆ g belongs to at most 6 triangles in Cay(G, S ), which is Condition (4). Indeed, if one edge e is labelled by s ∈ ∆ g , there are at most three possibilities to put an edge labelled by t ∈ ∆ g \ {s −1 } at each extremity of e, thus giving a maximum number of 2 · 3 = 6 triangles containing e. This also shows that for any s ∈ S we have We now turn our attention on the set ∆ g ∩ s∆ g . Its cardinality is equal to the number of pairs (u, v) ∈ ∆ g such that u = sv. By replacing u and v by the words g, g −1 , s −1 0 g and g −1 s 0 , this gives us 16 equations in the group. Among these 16 equations, 4 imply that s = 1. The 12 remaining equations for elements of ∆ g ∩ s∆ g are shown in Table 4.1.
In particular, if g is as in the conclusion of Proposition 4.4 for F = S ∪ s 0 S ∪ s −1 0 S and s = s 0 , we see that only the first two lines in this Table occur if C G (s 0 ) is locally finite, and only the first four occur otherwise. This implies Condition (5). Also, Condition 2 holds in this case since 1 belongs to s −1 0 S ⊆ F , and Condition (6) is automatically satisfied. The fact that there are infinitely many g in the conclusion of Proposition 4.4 imply that we can find such g satisfying also conditions (4.7)-(4.8).
We are now ready to prove Proposition 3.2. Possible elements of ∆ g ∩ s∆ g Occurs if Proof of Proposition 3.2. -The proof will be by successive applications of Lemma 4.7. To prove the proposition, it is enough that all elements of S belong to at least 7 S-triangles (to distinguish them from the newly added elements which will belong to at most 6 S-triangles) and that the numbers N 3 (s ±1 , S) for s in S are all distinct.
Let S 0 = S ∪ S −1 , so that |S 0 | 2|S|. Let s 1 , . . . , s n be any enumeration of the elements of S. Apply successively Lemma 4.7 at most 7 times with s 1 , to get a set S 1 containing S and such that N 3 (s 1 , S 1 ) is larger than 7. Then, applying Lemma 4.7 for s 2 at most 8 times, we can bring N 3 (s 2 , S 2 ) to another value 7. Doing the same for each element of S, we finally obtain a set S as in the lemma, after a total number of 7 + 8 + · · · + (|S| + 6) = |S|(|S| + 13)/2 successive applications of Lemma 4.7. At the end, we have S |S 0 | + 4 |S|(|S| + 13) 2 2|S|(|S| + 14).

Proofs of the main results
We collect here for completeness the straightforward proofs of the results from the introduction.
Proof of Theorem 1.1. -Let G be as in Theorem 1.1, and S 0 be a finite generating set. Let S 1 = (S 0 ∪ S 2 0 ∪ S 3 0 ) \ {1}, and S 2 be the generating set S given by Proposition 3.2 for S = S 1 . By Proposition 3.2 and the discussion preceding it, every automorphism of Cay(G, S 2 ) preserves the S 1 -colours: φ(gs) ∈ {φ(g)s, φ(g)s −1 } for every g ∈ G and s ∈ S 1 . By Theorem 3.1, φ is a left-translation by an element of G.
Proof of Corollary 1.2. -If G is finite, the equivalence is the content of [God81]. We can assume that G is infinite and finitely generated. The implication (2) =⇒ (1) is obvious, and the implication (1) =⇒ (3) is known and very easy, see [Wat71]. For the reader's convenience, we recall the argument. If an infinite finitely generated group G is either abelian or generalized dicyclic then there is a nontrivial permutation φ of G satisfying φ(gh) ∈ {φ(g)h, φ(g)h −1 } for every g, h ∈ G: take for φ the inverse map if G is abelian, and the map that is the identity on A and the inverse on G \ A if G is generalized dicyclic and x, A are as in Definition 2.1. In particular, φ induces an automorphism of every Cayley graph of G, different from a translation. Observe that this argument even rules out the existence of a non-locally finite GRR.
We have to justify (3) =⇒ (2). If G is not virtually abelian, then (2) is the conclusion of Theorem 1.1. Otherwise, G admits an element of infinite order (a torsion abelian finitely generated group is finite), and [LS21, Theorem 2] applies and proves (2).
Proof of Corollary

A conjecture on the squares of a random walk
We mentionned before Lemma 4.6 that we expect that the following conjecture holds.
Conjecture 6.1. -Let G be a finitely generated group that is not virtually abelian, µ be a symmetric probability measure on G with finite and generating support containing the identity, and (g n ) a realization of the random walk on G given by µ. Then   (6.1) ∀ a ∈ G, lim n P g 2 n = a = 0.
Better, there should be a function f : then G admits an abelian subgroup of index f (ε). This is known to be true for finite groups [Man18,Man94]. The main case of the conjecture is when a = 1: indeed with similar methods as the reduction to F = {1} in the proof of Proposition 4.4, one can show that the case a = 1 in Conjecture 6.1 implies the full conjecture for G not virtually 2-nilpotent.
Let us mention here that this conjecture would allow to greatly simplify our proofs, as it would imply immediately the following variant of Proposition 4.4, which also implies the main Theorem 1.1 by the same argument.
Lemma 6.2. -If Conjecture 6.1 holds for G, then for every s ∈ G and F ⊂ G finite there are infinitely many g ∈ G \ (sq −1 (F ) ∪ s sq −1 (F )) such that Proof. -We prove the stronger fact that the probability that g = g n satisfies the conclusion of the lemma is 1 − o(1) when |G : C G (s)| = ∞, and 1 |G: It follows from (6.1) that lim n P g n ∈ sq −1 (F ) = 0. It also implies that lim n P g n ∈ s sq −1 (F ) = 0. (6.2) To justify this, we need to introduce an independant copy (g n ) n 0 of the random walk (g n ). Since the support of µ is symmetric and generates G, there is a k such that P(g k = s −1 ) > 0. So using that g n+k is distributed as g k g n , we obtain P g n+k ∈ sq −1 (F ) P s −1 g n ∈ sq −1 (F ) and g k = s −1 This proves (6.2). Moreover, it follows from (4.5) that    lim n P (g −1 n sg n / ∈ F ) = 1 if |G : C G (s)| = ∞ lim n P (g −1 n sg n = s) = 1 |G:C G (s)| otherwise.
The conclusion follows.

Directed and oriented graphs
A natural variation of Cayley graphs is the concept of Cayley digraph (directed graph). Given a group G and a (not necessarily symmetric) generating set S ⊆ G\{1}, the Cayley digraph − − → Cay(G, S) is the digraph with vertex set G and with an arc (directed edge) from g to h if and only if g −1 h ∈ S.
A Cayley digraph − − → Cay(G, S) of G whose automorphism group acts freely on its vertex set is called a digraphical regular representation, or DRR. If moreover − − → Cay(G, S) has no bigons (that is if S ∩ S −1 = ∅) then we speak of an oriented graphical regular representation, or ORR.
We have the directed equivalent of Corollary 1.2: Proposition 7.1. -For a finitely generated group G, the following are equivalent: (1) G admits a DRR, (2) G admits a finite degree DRR, Proof.
-If G is finite, the equivalence is the content of [Bab80]. We can assume that G is infinite and finitely generated. The implication (2) =⇒ (1) is obvious, and the implication (1) =⇒ (3) is empty for infinite groups. We have to justify (3) =⇒ (2).
Let S be a finite generating set of G. Using Proposition 3.2 and [LS21, Lemma 32] we obtain a finite generating set S ⊆ S ⊂ G such that for all s ∈ S and t ∈ S, if N 3 (s, S) = N 3 (t, S) then t = s or t = s −1 . By [LS21, Lemma 5] and [LS21, Proposition 6] there exists a generating set T ⊆ S such that − − → Cay(G, T ) is a DRR. Moreover, T ∩ T −1 consist only of elements of order 2.
Observe that the equivalence of (1) and (3) was the content of [Bab78,Bab80]. We will conclude with the oriented equivalent of Corollary 1.2 and thus answer [Bab80, Problem 2.7]. Recall that a generalized dihedral group G is the semidirect product A Z/2Z where A is an abelian group and Z/2Z acts on A by inversion.
Proposition 7.2. -For a finitely generated group G, the following are equivalent: (1) G admits an ORR, (2) G admits a finite degree ORR, (3) G does not belong to the following list: • the non-trivial generalized dihedral groups, Proof. -If G is finite, the equivalence is the content of [MS18], while every generating set of a generalized dihedral group contains an element of order 2 (namely any element not in A). Once again, we have to justify (3) =⇒ (2) for G infinite.
Let G be a finitely generated group which is not generalized dihedral. Then by [Bab78, Proposition 5.2] there exists a finite generating set S of G without elements of order 2. Then the generating set S given by Proposition 3.2 and [LS21, Lemma 32] has also no elements of order 2. This implies that for T given by [LS21, Lemma 5] and [LS21, Proposition 6] the DRR − − → Cay(G, T ) is actually an ORR.