Aut-invariant quasimorphisms on free products

Let $G=A \ast B$ be a free product of freely indecomposable groups. We explicitly construct quasimorphisms on $G$ which are invariant with respect to all automorphisms of $G$. We also prove that the space of such quasimorphisms is infinite-dimensional whenever $G$ is not the infinite dihedral group. As an application we prove that an invariant analogue of stable commutator length recently introduced by Kawasaki and Kimura is non-trivial for these groups.


Introduction
The study of quasimorphisms on a given group G is an important branch of geometric group theory [Cal09] with quasimorphisms sharing deep relationships with the underlying structure of the group G. For free groups F n the so called counting quasimorphisms originating from the work of Brooks in [Bro81] yield a wide variety of examples. His ideas have been developed further by Calegari and Fujiwara who constructed unbounded quasimorphisms on non-elementary hyperbolic groups [CaFu10]. For diffeomorphism groups of surfaces many important constructions are given in [GaGh04]. There are also numerous applications in symplectic geometry originating from work of Entov and Polterovich [EnPo03]. Another fundamental paper on the geometry of quasimorphisms and central extensions is [BaGh92].
In this paper we construct unbounded Aut-invariant quasimorphisms on free products of groups. To achieve this we associate tuples of natural numbers we call codes to each element in a free product G = A * B. Inspired by Brooks counting quasimorphisms on free groups we then count these codes rather than actual elements of the free product and verify that this indeed yields quasimorphisms on G. We call them code quasimorphisms. We make use of an explicit description of the automorphism group of a free product found in [Gil87] to see in Proposition 4.11 that our code quasimorphisms are unbounded and invariant with respect to all automorphisms of G if A and B are not infinite cyclic.
If one of the factors of G = A * B is infinite cyclic, our code quasi-morphisms are not necessarily invariant under a specific class of automorphisms of G which is called the class of transvections. So, we slightly adjust the way we count codes for infinite cyclic factors and call the resulting maps weighted code quasimorphisms. We show in Proposition 5.7 that these are unbounded and invariant with respect to all automorphisms of G.
These two propositions together with an independent result for the free group on two generators from [BrMa19, Theorem 2] comprise the following result in Section 6 which is the main result of the paper.
Theorem 1. Let G = A * B be the free product of two non-trivial freely indecomposable groups A and B. Assume G is not the infinite dihedral group. Then G admits infinitely many linearly independent homogeneous Aut-invariant quasimorphisms, all of which vanish on single letters.
The infinite dihedral group does not admit any unbounded quasimorphism since all its elements are conjugate to their inverses. As a corollary of our construction we immediately deduce the existence of stably unbounded Aut-invariant norms on free products of two factors.
The following examples illustrate that the converse of Lemma 2.7 above is not true and finding unbounded Aut-invariant quasi-morphisms is much more difficult than finding unbounded Aut-invariant norms.
Example 2.8. Let Σ ∞ be the infinite symmetric group of finitely supported bijections of the natural numbers. The cardinality of the support defines an Aut-invariant norm of infinite diameter on Σ ∞ . However, any element g ∈ Σ ∞ has finite order. Therefore, no Aut-invariant norm on Σ ∞ is stably unbounded and any homogeneous quasi-morphism vanishes on all of Σ ∞ . Consequently, by Lemma 2.4 any quasimorphism on Σ ∞ is bounded.
Example 2.9. Let G = Z k for k ≥ 1. Since every g ∈ G lies in the same Aut(G)-orbit as g −1 , it follows that any homogeneous Aut-invariant quasimorphism vanishes on all of G. So any Aut-invariant quasimorphism is bounded on G. For k = 1 the standard absolute value defines a stably unbounded Aut-invariant norm on G, whereas for k ≥ 2 any Aut-invariant norm on G has finite diameter.
Example 2.10. Let G be the fundamental group of the Klein bottle G = Z * 2Z Z. Let a and b be generators of the two infinite cyclic factors of G in its above presentation. Consider the Aut-invariant word norm ν S generated by S = {ϕ(a ±1 ), ϕ(b ±1 ) | ϕ ∈ Aut(G)}. To see that this norm is unbounded on G we first note that commutator subgroup of G is a characteristic subgroup and G/[G, G] = Z/2 × Z, where Z/2 is a characteristic subgroup again. Consequently, the projection map p : G → Z sending p(a) = p(b) = 1 maps the set S to the Aut-invariant set {±1} in Z, which generates a stably unbounded Aut-invariant norm on Z. Therefore, the word norm ν S generated by S on G is stably unbounded as well.
However, there is no unbounded Aut-invariant quasimorphism on G. If ψ was such a quasimorphism, it could be chosen to be homogeneous by Lemma 2.4. Let ϕ be the automorphism inverting the generators a and b. Then ϕ inverts the center Z(G) = 2Z as well. Hence, ψ vanishes on Z(G). Similarly, every element of S = {(ab) n , (ba) n , (ab) n a, (ba) n b | n ∈ N} belongs to the same Aut-orbit that its inverse belongs to. So ψ vanishes on S as well. However, every element g ∈ G can be written as a product = zs where z ∈ Z(G) and s ∈ S. Therefore, ψ is bounded on all of G.
Definition 2.11. Let G = * i∈I G i be a free product of a family of groups {G i } i∈I for some indexing set I. For each i the factor G i is a subgroup of G via the canonical inclusion. An element of G that belongs to one of the factors is called a letter of G. Any product of letters is called a word in G. The product of any two letters belonging to the same factor in G can be replaced by the letter that represents their product in that factor. Moreover, any identity letters appearing in a word can be omitted without changing the element the word represents in G. Recall that any element g ∈ G has a unique presentation as a word, where no two consecutive letters lie in the same factor and no identity letters appear. Such a word is called reduced.
Lemma 2.12. Let I be a set of cardinality at least two. Let G i be a non-trivial group for all i and G = * i∈I G i be their free product. Let θ : G → R be a map whose absolute value is bounded on all letters of G by a constant B ≥ 0. Assume that there exists a constant D ≥ 0 such that |θ(w 1 w 2 ) − θ(w 1 ) − θ(w 2 )| ≤ D holds for all reduced words w 1 , w 2 for which their product w 1 w 2 is a reduced word. Then the map f : G → R defined by f (w) = θ(w) − θ(w −1 ) defines a quasimorphism of defect at most 12D + 6B, which is bounded on all letters by 2B.
Proof. Any element in G can be represented by a reduced word. So let w 1 , w 2 be reduced words. The word given by their product w 1 w 2 is reduced if and only if the last letter from w 1 belongs to a factor different from the one that the first letter of w 2 belongs to. Indeed, otherwise those two letters could be multiplied in their common factor and replaced by their product to shorten the number of letters appearing in the expression.
In order to bring w 1 · w 2 to its reduced form we first perform all cancellations which form a word we call c. After all cancellations have taken place the final potential reduction is to possibly replace a nontrivial product of two letters b and d belonging to the same factor by a non-trivial letter x representing their product in that factor. Therefore, we have two cases.
• The reduced presentations of w 1 and w 2 are given by w 1 = ac, w 2 = c −1 e and ae is the reduced presentation for w 1 · w 2 .
• The reduced presentations of w 1 and w 2 are given by w 1 = abc, w 2 = c −1 de, where b and d are letters belonging to the same factor. The reduced presentation of w 1 · w 2 is given by axe, where x = bd is the letter representing the non-trivial product of b and d.
We calculate for the second case that ≤6B + 12D.
The first case follows analogously. Since w 1 , w 2 were arbitrary reduced words and every element of G can be written in its reduced form, f is a quasimorphism of defect at most 6B + 12D. Since θ is bounded on all letters by B, so is f by 2B.
3 Aut-invariant quasimorphisms Definition 3.1. We call a group G freely indecomposable if G is non-trivial and not isomorphic to any free product of the form G 1 * G 2 where G 1 , G 2 are non-trivial groups.
Any free product of non-trivial groups has trivial center and contains elements of infinite order. So every abelian group and every finite group is freely indecomposable.
Lemma 3.2. Let ψ : G → R be a quasimorphism. Let {ϕ i } i∈I be a set of representatives for the elements of Out (G). If ψ is invariant under ϕ i for all i, then its homogenisationψ : G → R is invariant under all automorphisms of G.
Proof. The homogenisationψ is constant on conjugacy classes [Cal09,p.19]. By definitionψ is also invariant under the collection {ϕ i } i∈I , since ψ is. The result follows since any element ϕ ∈ Aut(G) can be written as the composition of some ϕ j with a conjugation.
Consider the free product G = G 1 * G 2 where G i is freely indecomposable for i = 1, 2. Following the exposition in [Gil87,p.116] based on results in [FoRa40] and [FoRa41] the automorphism group Aut(G 1 * G 2 ) is generated by the following types of automorphisms (1.-3.) if neither G 1 nor G 2 is infinite cyclic: 1. Elements from Aut(G 1 ) and Aut (G 2 ) give rise to automorphisms of G 1 * G 2 . These are called factor automorphisms.
2. Let g ∈ G i for some i ∈ {1, 2}. Define the map p g : G → G to be conjugation by g on the letters of G j for j = i and to be the identity on all letters from the group G i . This definition gives rise to an automorphism of G which is called a partial conjugation.
3. If G 1 ∼ = G 2 are isomorphic, interchanging the two factors is an automorphism of G. Such an automorphism is called a swap automorphism.
If G 1 ∼ = Z is infinite cyclic and the freely indecomposable group G 2 is not, then Aut(G 1 * G 2 ) is generated by the above automorphisms together with the following additional type of automorphisms: 4. Let s be a generator of G 1 and let a ∈ G 2 be any element. Then a transvection is the unique automorphism of G 1 * G 2 defined to be the identity on all letters from G 2 and maps s → as or s → sa.
Following the above description of the group of automorphisms of a free product of two factors we obtain: Lemma 3.3. Let G 1 , G 2 be freely indecomposable groups such that G 2 is not infinite cyclic. Then the outer automorphism group of their free product Out(G 1 * G 2 ) is generated by the images of Aut(G 1 ), Proof. By the universal property of the free product of two groups any automorphism is uniquely determined by its image on single letters. Let h ∈ G 1 and denote conjugation by h −1 on all of G by c h . Then Thus, p h and the factor automorphism given by conjugation by h −1 on G 1 represent the same element in Out (G). Similarly, in Out(G) partial conjugations on G 1 by elements from G 2 represent the same elements that factor automorphisms from G 2 do. Finally, any two choices of swap automorphism differ by a product of factor automorphisms.
Lemma 3.4. Let G be a group and H ≤ G be a characteristic subgroup with quotient projection p : G → G/H. Then for any unbounded Aut-invariant quasimorphism ψ : G/H → R the composition ψ • p : G → R is an unbounded Aut-invariant quasimorphism on G. Moreover, linearly independent quasimorphisms on G/H give rise to linearly independent quasimorphisms on G.
Proof. Clearly, ψ • p is a quasimorphism. The Aut-invariance of ψ • p on G follows from the Autinvariance of ψ on G/H together with the fact that H is characteristic. Finally, the statement about linear independence follows from the surjectivity of the projection to the quotient.

Code quasimorphisms
Recall that a tuple always refers to a finite sequence and so all tuples are naturally ordered.
Definition 4.1. Let A and B be groups. Write a given element g ∈ A * B in its reduced form. We assign two tuples of non-zero natural number that we will call codes as follows. Let (a 1 , . . . , a k ) be the tuple of letters from A appearing in the reduced form of g. We call (a 1 , . . . , a k ) the A-tuple of g. Then we count how often any one letter of (a 1 , . . . , a k ) appears consecutively. This yields a tuple of positive numbers A-code(g) = (n 1 , n 2 , . . . , n r ) which we call the A-code of g. Similarly, we obtain the B-tuple, which is the tuple of letters from B appearing in the reduced form of g, and the B-code of g, denoted B-code(g), by counting consecutive appearances of letters in the B-tuple.
Note that A-code(g) and B-code(g) might have very different length for elements g ∈ A * B in general.
Example 4.2. Let G = A * B where A = Z/5 and B is any group. Let a ∈ A, b ∈ B be nontrivial elements. Consider g = a 2 bababa 4 baba. The A-tuple of g is (a 2 , a, a, a 4 , a, a) and therefore A-code(g) = (1, 2, 1, 2). However, the B-tuple of g is (b, b, b, b, b) and so B-code(g) = (5).
Remark 4.3. The code of any element g ∈ A * B is clearly invariant under all factor automorphisms.
The following lemma is immediate.
In the spirit of Brooks counting quasimorphisms we will now define code quasimorphisms, which are counting the occurrences of a string of natural numbers in the A-code and B-code associated to an element in the free product A * B.
Definition 4.5 (Code quasimorphisms). Let k ≥ 1 and let z = (n 1 , . . . , n k ) be a tuple of non-zero natural numbers n 1 , . . . , n k for some k ∈ N. Let C ∈ {A, B}. Define θ C z : A * B → Z ≥0 to count the maximal number of disjoint appearances of z as a tuple of consecutive numbers in the C-code for all g ∈ A * B. Further, define the code quasimorphism Example 4.6. Let G = Z/5 * B and g = a 2 bababa 4 baba for non-trivial a ∈ A, b ∈ B as in Example 4.2. For z = (1, 2) we calculate θ A z (g) = 2 and θ A z (g −1 ) = θ Ā z (g) = 1 and so f A z (g) = 2 − 1 = 1.
Lemma 4.8. Let A, B be non-trivial groups and let C ∈ {A, B}. For a non-empty tuple of non-zero natural numbers z the map f C z : A * B → Z defines a quasimorphism that is bounded on letters and invariant with respect to all factor automorphisms. Moreover, D(f C z ) ≤ 30.
Proof. We want to apply Lemma 2.12 to deduce that f C z is a quasimorphism. Clearly, |θ C z (x)| ≤ 1 for all letters x ∈ A * B and all z. Let w 1 , w 2 be reduced words representing elements in A * B such that their product w 1 w 2 is reduced. That is, the last letter of w 1 and the first letter of w 2 belong to different factors. Without loss of generality we can assume C = A. Let A-code(w 1 ) = (n 1 , . . . , n k ) and A-code(w 2 ) = (m 1 , . . . , m ℓ ). Let x be the last letter from A in w 1 and let y be the first letter from A in w 2 . Then since at most one of the disjoint occurrences of z can involve numbers that do not lie completely in the A-code of either w 1 or w 2 . If x = y, then θ C z (w 1 w 2 ) ≥ θ C z (w 1 ) + θ C z (w 2 ) − 2 since n k and m 1 can each be contained in at most one occurrences of z in the A-code of w 1 and w 2 . Moreover, if an occurrence of z in the A-code of w 1 w 2 involves n k + m 1 , then all other occurrences are fully contained in either the A-code of w 1 or w 2 . Thus, θ C z (w 1 w 2 ) ≤ θ C z (w 1 ) + θ C z (w 2 ) + 1. In both cases we conclude and it follows from Lemma 2.12 that f C z is a quasimorphism of defect D(f C z ) ≤ 30. Moreover, by Remark 4.3 the maps θ C z are invariant under all factor automorphisms of A * B. Consequently, f C z = θ C z − θ C z is invariant under factor automorphisms as well.
Proposition 4.11. Let A * B be a free product of two freely indecomposable groups A and B, neither of which is infinite cyclic. Then for any generic tuple of natural numbers z the following holds: In both cases the space of homogeneous Aut-invariant quasimorphisms on A * B that vanish on letters has infinite dimension.
Proof. First, consider the case A ≇ B. Since A * B is not the infinite dihedral group, at least one of the factors is not isomorphic to Z/2. Without loss of generality we assume A ≇ Z/2. Let z be generic. By Lemma 4.8 the map f A z defines a quasimorphism invariant under all factor automorphisms. According to Lemma 3.3 this means that f A z is invariant under a full set of representatives for Out(A * B). Therefore, the homogenisationf A z is invariant under all automorphisms of A * B by Lemma 3.2. It remains to check thatf A z is unbounded, which is equivalent to checking that f A z itself is unbounded by Lemma 2.4. Since A ≇ Z/2, it satisfies |A| ≥ 3 and we can choose two distinct non-trivial elements a 1 , a 2 ∈ A. Furthermore, choose a non-trivial element b ∈ B. Let z = (n 1 , . . . , n k ) and choose m ∈ N to be non-zero and distinct from all n i ∈ N. We set The A-code of w is given by A-code(w) = (n 1 , . . . , n k ) = z if k is even, (n 1 , . . . , n k , m) = (z, m) if k is odd.
Since w starts and ends with letters from different groups, the reduced expression of w ℓ is the ℓ-fold product of the word w for all ℓ ∈ N. Moreover, because the first letter from A in w is a 1 and the last letter from A is a 2 , the A-code of w ℓ is A-code(w ℓ ) = (z, z, . . . , z) if k is even, (z, m, z, m, . . . , z, m) if k is odd.
Since m is distinct from all n i , m can never appear in any occurrence of z orz in the A-code of w ℓ . So θ A z (w ℓ ) = ℓ, whereas θ Ā z (w ℓ ) = 0 since z is generic. Consequently, which shows that f A z is unbounded. Second, consider the case A ∼ = B and fix a choice of isomorphism. Let z be generic. It holds that |A| = |B| ≥ 3 since A * B is not the infinite dihedral group. Consider the swap isomorphism s interchanging the factors A and B, where we use the fixed isomorphism from before to identify A and B with each other. Then the application of s to any element g interchanges the A-code and B-code of g with each other. This implies that the sum θ A z + θ B z is invariant under s and consequently the sum For this let a 1 , a 2 ∈ A and b 1 , b 2 ∈ B be non-trivial such that a 1 = a 2 and b 1 = b 2 . Pick a non-zero number m ∈ N distinct from all n i ∈ N, where z = (n 1 , . . . , n k ). As before, we set where s is 1 or 2 depending on whether k is odd or even. We set Then the A-code and B-code of w agree and are given by A-code(w) = B-code(w) = (n 1 , . . . , n k ) = z if k is even, (n 1 , . . . , n k , m) = (z, m) if k is odd.
Since m is distinct from all n i , m can never appear in any occurrence of z orz in the A-code and B-code of w ℓ . As in the first case, θ A z (w ℓ ) = θ B z (w ℓ ) = ℓ, whereas θ Ā z (w ℓ ) = θ B z (w ℓ ) = 0 since z is generic. Consequently, which shows that f A z + f B z is unbounded and therefore its homogenisation is the desired unbounded Aut-invariant quasimorphism on A * B.
Finally, let us verify that the space of homogeneous Aut-invariant quasimorphisms on A * B that vanish on letters is infinite-dimensional. Let r ∈ N and let z 1 , . . . , z r be generic tuples. Choose z r+1 be a 3-tuple whose entries are distinct non-zero natural numbers and do not appear in any of the z i ; then z r+1 is generic. It follows from the above construction of the word w for z r+1 in both cases that any linear combination of the associated quasimorphisms f A z 1 + f B z 1 , . . . , f A zr + f B zr vanishes on all powers of w. It follows that the same holds for any linear combination of their homogenisationsf A z 1 +f B z 1 , . . . ,f A zr +f B zr . Thus,f A z r+1 +f B z r1 is not contained in the subspace spanned by the first r quasimorphisms. Clearly, the homogenisation of any code quasi-morphism vanishes on all letters of A * B. Since r ∈ N was arbitrary, it follows that the space of homogeneous Aut-invariant quasimorphisms on A * B that vanish on letters cannot have finite dimension.

Weighted code quasimorphisms
If one of the factors of a free product A * B of freely indecomposable groups happens to be infinite cyclic, the code quasimorphisms above are in general not Aut-invariant since they are not necessarily invariant with respect to transvections. Thus, we need to modify our original construction to deal with infinite cyclic factors. Afterwards we will follow steps similar to the previous section in order to establish their Aut-invariance.
Lemma 5.1. Let B be a non-trivial group and let w be any word in Z * B such that w only contains letters of the same sign from Z and starts and ends with a non-zero letter from Z. Then its unique reduced form w ′ starts and ends with a letter from Z with that given sign. Moreover, the sum over all letters in w belonging to the factor Z remains the same in its reduced form w ′ .
Proof. Any word in the free product is brought to its reduced form by successively eliminating trivial letters and replacing two adjacent letters from the same factor by their product in that factor. The sum of all letters from Z stays the same because any two adjacent letters of Z are always replaced by their sum throughout the reduction process. The only way to encounter an elimination of the first letter a 1 ∈ Z or the last letter a n ∈ Z during the reduction process would be by the occurrence of −a 1 or −a n . This is not possible since a 1 and a n are non-zero and all letters have the same sign by assumption.
Definition 5.2 (Weighted Z-code). Let B be freely indecomposable and B ≇ Z. Write g ∈ Z * B in reduced form. Let (a 1 , . . . , a k ) be the Z-tuple of g. We define a tuple (x 1 , . . . , x ℓ ) of non-zero natural numbers as follows. Consider the successive subsequences of maximal length in (a 1 , . . . , a k ) consisting of integers all of the same sign. For the i-th such sequence, we define x i to be the absolute value of the sum of integers in that sequence. We call the tuple (x 1 , . . . , x ℓ ) the weighted Z-code of g.
since at most one of the disjoint occurrences of z can involve numbers that do not lie completely in the weighted Z-code of either w 1 or w 2 .
If sgn(n k ) = sgn(m 1 ), then θ Z z (w 1 w 2 ) ≥ θ Z z (w 1 ) + θ Z z (w 2 ) − 2 since only one occurrence of z in the weighted Z-code of w 1 and w 2 can involve the first or last number respectively. Moreover, if an occurrence of z in the weighted Z-code of w 1 w 2 involves x k ′ + y 1 , then all other occurrences are fully contained in the weighted Z-code of either w 1 or w 2 . Thus, θ Z z (w 1 w 2 ) ≤ θ Z z (w 1 ) + θ Z z (w 2 ) + 1. In both cases we conclude that It follows from Lemma 2.12 that f Z z is a quasimorphism of defect at most 30.
Lemma 5.6. For all non-empty tuples z the weighted code quasimorphism f Z z : Z * B → Z is invariant under factor automorphisms and transvections.
Proof. It is immediate from the definition that the weighted Z-code of any element in the free product is invariant under factor automorphisms. Let x be a generator of the infinite cyclic factor in Z * B. Any transvection is defined to be the identity on letters from B and maps x → xy or x → yx for some nontrivial element y ∈ B. Let us consider the transvection ϕ uniquely specified by x → xy and show that the weighted Z-code of any element in Z * B is invariant under ϕ. Then it immediately follows that θ Z z and f Z z are invariant under ϕ. The argument for transvections of the second kind will follow analogously to the one we present now.
Let w ∈ Z * B be a reduced word such that its weighted Z-code has length one. This means that all letters from Z in the reduced expression of w have the same sign and the weighted Z-code is given by the image of w under the factor projection Z * B → Z. Note that this factor projection is invariant with respect to ϕ and so the weighted Z-code of ϕ(w) agrees with the one of w. There cannot be any cancellations of letters from Z occurring.
Let us do a preliminary calculation to visualise the general case more easily. Let k, ℓ be non-zero natural numbers and b ∈ B non-trivial. Then This shows that the letter from B separating the positive and negative powers of x either remains b or is a conjugate of b in B after applying ϕ.
Let w ∈ Z * B be a reduced word with weighted Z-code of length k ≥ 2. In w we formally gather all consecutive occurrences of powers of x of the same sign and call these sub-words w i for i = {1, . . . , k}. That is, we write the reduced word w uniquely as a product of reduced words as where the b i ∈ B are non-trivial and the w i are of of maximal length such that all letters from Z inside any w i have the same sign. Moreover, in this decomposition w 1 ends with a letter from Z, w n starts with a letter from Z and all other w i start and end with letters from Z. By the maximality of w i all letters from Z occurring in w i have different signs from the ones occurring in w i+1 for all i.
We apply ϕ to w and obtain an a priori not necessarily reduced word, which we rewrite in the previous block form as where b ′ i = yby −1 if the letters from Z change sign from positive to negative at b i and b ′ i = b i if they change from negative to positive. Moreover, all letters from Z inside any w ′ i have the same sign again, w ′ 1 ends with a letter from Z, w ′ n starts with a letter from Z and all other w ′ i start and end with letters from Z.
We observe that when bringing ϕ(w) to its reduced form there cannot be any cancellations of the letters b ′ i . This is because by Lemma 5.1 the letters that are adjacent to b i will always remain letters from Z after the reduction procedure of all w ′ i . Indeed, replacing all w ′ i by their reduced forms w ′′ i we see that the product is the reduced representative of ϕ(w) since the letters adjacent to the b ′ i are always letters from Z. Consequently, no cancellations in between letters of different signs from Z can occur when bringing ϕ(w) to its reduced form. The reduced words w ′′ i have the same weighted Z-code as the original w i for all i. Therefore, the weighted Z-code of ϕ(w) agrees with the weighted Z-code of w.
which shows that f Z z is unbounded. Finally, let us verify that the space of homogeneous Aut-invariant quasimorphisms on Z * B that vanish on letters is infinite-dimensional. Let r ∈ N and let z 1 , . . . , z r be generic tuples. Choose z r+1 be a 3-tuple whose entries are distinct non-zero natural numbers and do not appear in any of the z i ; then z r+1 is generic. It follows from the above construction of the word w for z r+1 that any linear combination of f Z z 1 , . . . , f Z zr vanishes on all powers of this w. It follows that the same holds for any linear combination of their homogenisationsf Z z 1 , . . . ,f Z zr . Thus,f Z z r+1 is not contained in the subspace spanned by the first r quasimorphisms. Clearly, the homogenisation of any weighted code quasi-morphism vanishes on all letters of Z * B. Since, r ∈ N was arbitrary, it follows that the space of homogeneous Aut-invariant quasimorphisms on Z * B that vanish on letters cannot have finite dimension.

Applications of code quasimorphisms
Proof of Theorem 1. By [BrMa19, Theorem 2] the space of homogeneous Aut-invariant quasimorphisms on Z * Z is infinite-dimensional. Inverting both generators of the factors defines an automorphism which inverts all letters in Z * Z. So any homogeneous Aut-invariant quasimorphism on Z * Z vanishes on all letters. For all other free products of two factors Proposition 4.11 and Proposition 5.7 imply the existence of infinitely many linearly independent homogeneous Aut-invariant quasimorphisms, all of which vanish on letters.
Proof of Corollary 1.1. Let A * B be a free product of two freely indecomposable groups which is not the infinite dihedral group. By Theorem 1 there exist unbounded Aut-invariant quasimorphisms on A * B that are bounded on all letters. Since A * B is generated by letters, the result follows from Lemma 2.7.
Remark 6.1. If neither A nor B is infinite cyclic, then Corollary 1.1 can also be deduced from the result given in [Mar20,Lemma 4.4] together with the explicit description of the automorphism group given in Section 3.
Corollary 6.2. Let G = * i∈I G i be a free product of finitely many freely indecomposable groups G i . Assume there exist free factors G j and G k with j = k such that no free factors G i for i / ∈ {j, k} is isomorphic to G j or G k or is infinite cyclic. Moreover, assume that G j , G k are not both equal to Z/2. Then any unbounded Aut-invariant quasimorphism on G j * G k gives rise to an unbounded Aut-invariant quasimorphism on G. In particular, the space of homogeneous Aut-invariant quasimorphisms on G is infinite-dimensional.
Proof. We claim that the projection p : G → G j * G k is Aut-equivariant, i.e. any automorphism of G descends via p to an automorphism of G j * G k . This is equivalent to ker(p) being a characteristic subgroup of G. Once this is established, we apply Theorem 1 to G j * G k and conclude the proof by applying Lemma 3.4.
Let us now show that any automorphism of G indeed descends to G j * G k . By [Gil87] Aut(G) is generated by factor automorphisms, swap automorphisms, partial conjugations and transvections since inner automorphisms can be written as products of factor automorphisms and partial conjugations. It is clear that all factor automorphisms and all partial conjugations of G descend to automorphisms of G j * G k via p. By our assumption there are no swap automorphisms permuting any other free factors in G with G j and G k , so these descend to the quotient as well. It only remains to check the transvections if G j or G k happen to be infinite cyclic. So let G j be infinite cyclic generated by x and let a be a letter from a different factor G ℓ . If ℓ = k, then any transvection ϕ a defined by ϕ a (x) = ax or ϕ a (x) = xa descends via p to the same transvection on G j * G k . If ℓ = k, any such transvection descends to the identity on G j * G k . In particular, it always descends via p. Since a generating set of Aut(G) descends to automorphisms of the quotient G j * G k , any element of Aut(G) does so. Consequently, the map p is Aut-equivariant. Corollary 6.3. Let H → G → A * B be an extension of a free product of freely indecomposable groups A and B by a group H. Assume that H is a characteristic subgroup of G and A * B is not the infinite dihedral group. Then the space of homogeneous Aut-invariant quasimorphisms on G is infinitedimensional.
Proof. The space of homogeneous Aut-invariant quasimorphisms on A * B is infinite-dimensional by Theorem 1. Therefore, the result follows from Lemma 3.4.
Corollary 6.4. Let G 1 * H G 2 be a free product of groups G 1 , G 2 amalgamated over a common subgroup H which is proper and central in both G 1 and G 2 . If G 1 /H and G 2 /H are freely indecomposable and not both equal to Z/2, the space of homogeneous Aut-invariant quasimorphisms on G 1 * H G 2 is infinite-dimensional.
Proof. By assumption H = G 1 and H = G 2 and so H equals the center of G 1 * H G 2 . As such it is a characteristic subgroup of H is not isomorphic to the infinite dihedral group and the result follows from Corollary 6.3 above.
Example 6.8. Let G p,q = Z * Z Z be the free product of two copies of the integers amalgamated over inclusions ι 1 , ι 2 : Z → Z which are multiplication by p and q. For coprime choices of p and q these are the so called knot groups K p,q arising as the fundamental group of the complement of torus knots. Then G p,q admits infinitely many linearly independent homogeneous Aut-invariant quasimorphisms if min{|p|, |q|} ≥ 2 and max{|p|, |q|} ≥ 3. We have seen in Example 2.10 that this is no longer true for p = q = 2. Example 6.9. B 3 * Z B 3 , the free product of B 3 with itself amalgamated over their common center generated by the Garside element, admits infinitely many linearly independent unbounded Aut-invariant quasimorphisms. To prove this we cannot apply Corollary 6.4 directly since B 3 /Z is not freely indecomposable. The center of B 3 * Z B 3 is again generated by the Garside element of each of the factors. This fits into the short exact sequence Finally, (B 3 /Z) * (B 3 /Z) = PSL(2, Z) * PSL(2, Z) = Z/2 * Z/3 * Z/2 * Z/3. So Corollary 6.2 applies with G j = G k = Z/3. Then the statement for B 3 * Z B 3 follows from Lemma 3.4.

Aut-invariant stable commutator length
For any group G let cl G denote the commutator length on [G, G], which is defined to be the minimal number of commutators required to write a given element of the commutator subgroup. Let scl G (x) = lim n cl(x n ) n denote the stable commutator length of x ∈ [G, G]. It shares a deep relationship with quasimorphisms on G through the so called Bavard duality [Cal09]. We now define the Autinvariant (stable) commutator length; this is a special case of theĜ-invariant (stable) commutator length defined in [KaKi20]. Then for x ∈ [Ĝ, G] thê G-invariant commutator length clĜ ,G (x) is defined to be the minimal length of an expression of x as a product of commutators [F, g] and their inverses where F ∈Ĝ and g ∈ G. TheĜ-invariant stable commutator length sclĜ ,G for x ∈ [Ĝ, G] is defined by sclĜ ,G (x) = lim n clĜ G (x n ) n . Given any group G, its inner automorphism group Inn(G) is a normal subgroup of Aut(G) and so the above definition applies to Inn (G). If G has trivial center, G can be identified with Inn (G). In this case we simplify the notation by denoting the Aut(G)-invariant commutator length simply as cl Aut and the Aut(G)-invariant stable commutator length simply as scl Aut . SettingĜ = Aut(G) the following lemma is proven in [KaKi20, Lemma 2.1].
In fact, according to [KaKi20, Theorem 1.3]Ĝ-invariant quasimorphisms satisfy an analogue of the Bavard duality theorem if [Ĝ, G] = G. All free products A * B of freely indecomposable groups A and B have trivial center and so the notions cl Aut and scl Aut apply. However, free products often fail to satisfy [Aut (G), G] = G. We will use a constructive approach rather than relying on an invariant analogue of Bavard's duality in the following. Example 7.4. Consider G = PSL(2, Z) = Z/3 * Z/2. Then Aut(G) is generated by the set C consisting of the non-trivial factor automorphism of Z/3 and conjugations by letters of Z/3 and Z/2, since both free factors are abelian groups. Consequently, [Aut(G), G] is normally generated by commutators of the form [c, g] = cgc −1 g −1 = c(g)g −1 for c ∈ C and g ∈ G. In all expressions [c, g] the letter b representing the non-trivial element of the factor Z/2 arises an even number of times. Therefore, b / ∈ [Aut(G), G] and the latter is not the full group G.
Lemma 7.5. Let G = A * B be a free product of freely indecomposable groups where at least one of the factors is infinite cyclic. Then [Aut (G), G] has index at most 2 in G. Therefore, any unbounded quasimorphism on G is unbounded when restricted to [Aut(G), G].
Proof. If an unbounded quasimorphism q is bounded on a finite index subgroup H ≤ G, its homogenisationq vanishes on H. Thenq descends to a map of sets on the finite set G/H implying that the image of q is bounded and so was the image of q to begin with. Therefore, any quasimorphism q with unbounded image cannot be bounded on H. So, it remains to show that the index of [Aut (G), G] in G is finite to prove the lemma.
First, consider the case where A and B are both infinite cyclic and so G can be identified with F 2 , the free group of rank 2. Let x and y be standard generators. Consider the automorphism ϕ of F 2 defined by ϕ(x) = yx and ϕ(y) = y. Then [ϕ, x] = ϕ(x) · x −1 = y. So y ≤ [Aut(F 2 ), F 2 ]. By symmetry of the generating set it holds that x ≤ [Aut(F 2 ), F 2 ] as well and it follows that [Aut(F 2 ), F 2 ] = F 2 .