On Bounded Cohomology of Amalgamated Products of Groups

We investigate the structure of the singular part of the second bounded cohomology group of amalgamated products of groups by constructing an analog of the initial segment of the Mayer-Vietoris exact cohomology sequence for the spaces of pseudocharacters.


INTRODUCTION
We recall that bounded cohomology H * b (G) of a group G (we will be considering only cohomology with coefficients in the additive group of reals with trivial action, so in our notations for cohomology the coefficient module will be omitted) is defined using the complex of bounded cochains f : G × · · · × G → , and δ n b = δ n | C n b (G) is the bounded differential operator. Since H 0 b (G) = and H 1 b (G) = 0 for any group G, investigation of bounded cohomology starts in dimension 2. One observes that H 2 b (G) contains a subspace H 2 b ,2 (G) (called the singular part of the second bounded cohomology group), which has a simple algebraic description in terms of quasicharacters and pseudocharacters, and the quotient space H 2 b (G)/H 2 b ,2 (G) is canonically isomorphic to the bounded part of the ordinary cohomology group H 2 (G). See [6] for background and available results on bounded cohomology of groups. (For bounded cohomology of topological spaces see [8]. ) We recall that a function F : G → is called a quasicharacter if there exists a constant C F 0 such that |F (x y) − F (x) − F (y)| C F for all x, y ∈ G.
A function f : G → is called a pseudocharacter if f is a quasicharacter and in addition f (g n ) = n f (g ) for all g ∈ G and n ∈ .
The notions of a quasicharacter and a pseudocharacter originally arose from the questions of stability of solutions of functional equations [9,10,11] and continuous representations of groups [12]. We use the following notation: • X (G) = the space of additive characters G → ; • QX (G) = the space of quasicharacters; • P X (G) = the space of pseudocharacters; • B(G) = the space of bounded functions.
Special interest in H 2 b ,2 is motivated in part by its connections with other structural properties of groups such as commutator length [1] and bounded generation [6]. (See [3] for a simple proof of triviality of H 2 b ,2 for Chevalley groups over rings of S-integers in algebraic number fields using bounded generation.) For example, Grigorchuk proved [7] (cf. also [5]) that the amalgamated product A 1 * H A 2 does not have bounded generation provided that the number of double cosets of A 1 modulo H is at least 3 and [A 2 : H ] 2 by showing that dim H 2 b ,2 (A 1 * H A 2 ) = ∞ in this case. The proof is based on the explicit construction (see Example 4.4) of an infinite family of linearly independent quasicharacters which naturally generalize the construction of quasicharacters for free groups. The quasicharacters for free groups were first constructed by Brooks [2], and Faiziev showed that they can be used to find a basis for the space of pseudocharacters of a free group [4] (cf. [6,Theorem 5.7] for a shorter and more conceptual proof).
However, no systematic study of bounded cohomology of amalgamated products of groups has been undertaken. The goal of this paper is to provide the first step in an attempt to obtain general information about bounded cohomology of amalgamated products of groups. Since the main technical tool used to compute cohomology of the amalgamated product A 1 * H A 2 is the Mayer-Vietoris exact sequence (see [14, The- it is natural to try to exhibit an analog of this sequence for bounded cohomology. We construct an initial segment of this sequence for bounded cohomology, it starts in dimension 2, and we formulate our results in terms of spaces of pseudocharacters. We begin by considering the case when the amalgamated subgroup is normal in both factors (Theorem 2.1). In the general case we restrict our attention to the special class of pseudocharacters which we call H -spherical (Theorem 4.6); see §4 for relevant definitions and discussion.
The sequences (2) and (13) constructed in Theorems 2 and 5, respectively, reduce the problem of computation of spaces of pseudocharacters for amalgamated products of groups to that for free products of groups (terms on the left); the structure of the latter space is known [6,Proposition 4.3 and Remark 4.4].
We conclude this section with two easy facts which will be used throughout the paper without special reference. Lemma 1.1 Any pseudocharacter is constant on conjugacy classes; a bounded pseudocharacter is trivial.
Proof. Let f ∈ P X (G) and suppose that f (y x y −1 ) − f (x) = a = 0 for some x, y ∈ G.
Then the difference f (y x n y −1 ) − f (x n ) = na is unbounded when n → ∞. On the other hand a contradiction. The second assertion is obvious.

THE CASE OF A NORMAL SUBGROUP
In this section we will establish an analog of the initial segment of the Mayer-Vietoris sequence for spaces of pseudocharacters assuming that the amalgamated subgroup N is normal in both factors A 1 and A 2 (in which case it is also normal in the amalgamated product). To describe this sequence we need to introduce some natural linear maps. First, we define is the restriction map associated with the natural embedding In contrast to the usual Mayer-Vietoris sequence for the spaces of characters whereβ andγ are analogous to β and γ introduced above, the sequence for pseudocharacters will contain at the extreme left one extra term which is typically an infinite dimensional vector space. To define it, we consider the embedding α: induced by the natural surjective homomorphism and let P X 0 (A 1 /N ) * (A 2 /N ) denote the kernel of the linear map β: is the restriction map induced by the natural embedding The above spaces and linear maps align in the following sequence.
Theorem 2.1 Let N be a normal subgroup of A 1 and A 2 . Then the sequence of vector spaces is exact.
We will prove the theorem in the next section, and now will derive two consequences.

Corollary 2.2
Given two arbitrary pseudocharacters f 1 and f 2 on the groups A 1 and A 2 respectively, there exists a pseudocharacter f on the free product A 1 * A 2 such that f | A i = f i , i = 1, 2. Corollary 2.3 Let A be an arbitrary group, N be its normal subgroup. Then the restriction homomorphism ρ: P X (A * N A) → P X (A) induced by embedding A into A * N A as either factor, is surjective. If, moreover, [A : N ] = 2 then ρ is an isomorphism.
Proof. For the first assertion, one needs to observe that for any f ∈ P X (A), the pair ( f , f ) belongs to Ker γ , and therefore is obtained as the restriction of a pseudocharacter on A * N A. If 3.1. Exactness in the term P X (A 1 ) ⊕ P X (A 2 ). The inclusion Im β ⊆ Ker γ being obvious, all we need to prove is that given pseudocharacters The following observation saves the (serious) trouble of verifying the condition f (g n ) = n f (g ).
Lemma 3.1 In the current notation, for the existence of pseudocharacter f it suffices to construct a quasicharacter F ∈ QX (A 1 * N A 2 ) such that the differences F | A i − f i are bounded for i = 1, 2.
Proof. Indeed, it follows from (1) that given such an F , there exists a pseudocharacter f ∈ P X (A 1 * N A 2 ) for which the difference F − f is bounded. Then for i = 1, 2, the difference The construction of such a quasicharacter F ∈ QX (A 1 * N A 2 ) rests on a specific choice of systems of representatives X i of all left cosets = N in A i /N for i = 1, 2. Namely, it is possible to choose such systems of representatives X i having the following property: Indeed, letS i denote the set of elements of order two in A i /N , and pick an arbitrary system of representatives S i ⊆ A i of the cosets fromS i . Since for each Choose an arbitrary system of representatives T i ⊆ A i of the cosets fromT i , and let Suppose that the systems of representatives X i with the property (3) have been chosen. We define an involutive transformation τ i : X i → X i by setting Now, we let X = X 1 ∪ X 2 (disjoint union), and introduce a function F and an involution τ on X whose restrictions to X i are f i and τ i respectively: Let W be the set of all words of the form x 1 · · · x n where x i ∈ X and for every i = 1, . . . , n − 1, the elements x i and x i+1 belong to different parts X 1 or X 2 of X (by convention, the empty word is included in W and corresponds to n = 0). Then each element g ∈ G := A 1 * N A 2 admits a unique canonical presentation of the form for some h ∈ N and some word x 1 · · · x n ∈ W (cf., for example, [13, Chapter I, Theorem 1]). Using the canonical form (4), we can extend τ to an involutive transformation of G by setting Let f 0 ∈ P X (N ) denote the common restriction of f 1 and f 2 to N : It follows from (5) that for every g ∈ G we have g τ(g ) ∈ N , so the expression f 0 (g τ(g )) makes sense. Let S = S 1 ∪ S 2 and T = T 1 ∪ T 2 . We extend F to a function on A 1 * N A 2 by the formula and, by convention, η(g ) = 0 if g ∈ N (in this definition we use the unique canonical presentation (4)).
First, we observe that for h ∈ N we have In particular, F | N = µ| N is a pseudocharacter with constant C = C f 0 . Also, writing g ∈ G in the canonical form and using the fact that f 0 is the common restriction of pseudocharacters f 1 and f 2 to N , we obtain Next, we will show that the difference In particular, we obtain To complete the proof of the proposition it remains to show that F is a quasicharacter on the entire amalgamated product A 1 * N A 2 . For a function f on G we define So, we need to show that δF = δµ + δη is bounded on G × G. For convenience of further reference, we will collect in the following lemma some properties of the functions µ and η.
and (i) follows. For (ii), one needs to observe that Properties (iii)-(v) follow immediately from the definition of η.
For given two elements g 1 , g 2 ∈ G we pick the canonical presentations g 1 = x 1 · · · x m h 1 and g 2 = y 1 · · · y n h 2 where x 1 · · · x m , y 1 · · · y n ∈ W and h 1 , h 2 ∈ N . We first consider the easiest case when x m and y 1 belong to different factors A i , i = 1, 2. In this case the canonical presentation of g 1 g 2 is Since τ(x 1 · · · x m y 1 · · · y n ) = τ(y 1 · · · y n )τ(x 1 · · · x m ) we conclude from Lemma 3.3 (i,ii) and (7) that On the other hand, It follows that in this case To consider the general case we need to introduce the fragments of g 1 and g 2 that cancel out in g 1 g 2 . Let k be the largest integer min{m, n} such that x m−i+1 y i ∈ N for all i = 1, . . . , k. We introduce the following elements and v 2 = y 1 · · · y k , u 2 = y k+1 , w 2 = y k+2 · · · y n , where by convention v 1 = e if k = 0, w 1 = e if m = k + 1, and w 1 = u 1 = e if m = k, with similar rules for v 2 , u 2 , and w 2 . We observe that v 2 = τ(v 1 ), so letting v = v 1 we have the following factorizations: It follows from our construction that both u 1 and u 2 belong to the same factor A i , so we can write u 1 u 2 = z h for some h ∈ N , z ∈ X . We claim that This estimation is a consequence of the following three inequalities that reflect the three-step transition from g 1 , g 2 to u 1 , u 2 .
3.2. Exactness in the term P X (A 1 * N A 2 ). The exactness of (2) in P X (A 1 * N A 2 ) is based on the following fact.
Lemma 3.5 Let G be an arbitrary group, N be its normal subgroup. If a pseudocharacter f ∈ P X (G) has zero restriction to N , then it satisfies f (g h) = f (g ) for all h ∈ N , g ∈ G.
In other words, the natural sequence On the other hand, (g h) n can be written as g n h for h ∈ N , so If f ∈ Ker β, then f | N = 0. Lemma 3.5 implies that f factors through the group homomorphism immediately implying that f ∈ Im α and proving the inclusion Ker β ⊆ Im α. The opposite inclusion is obvious.

3.3.
Remarks. The general construction of a quasicharacter F ∈ QX (A 1 * N A 2 ) lifting given pseudocharacters f i ∈ P X (A i ), (i = 1, 2) with the same restrictions to N essentially simplifies in the following two particular cases: (1) S = ∅, i.e., when the quotients A 1 /N and A 2 /N do not have elements of order two; (2) T = ∅, i.e., when these quotients are groups of exponent two. In the first case with the above choice of the coset representative systems F can be extended "by linearity":

THE CASE OF AN ARBITRARY SUBGROUP
If we do not assume that the amalgamated subgroup H is normal in both factors A 1 and A 2 , then two difficulties arise. First of all, when we switch representatives of cosets modulo H with elements of H in order to write a product of two words in the canonical form, the representative of a coset will change. Secondly, there is no natural candidate for the term on the extreme left. We restrict our attention to special classes of quasicharacters and pseudocharacters which we call strongly H -spherical and H -spherical respectively. We would like to point out that the only explicitly known quasicharacters on amalgamated products (see Example 4.4) are strongly H -spherical. Below is a brief analysis of what restrictions should be imposed on pseudocharacters.
Let H be a subgroup of a group G. The first conjecture, that naturally arises after preliminary considerations, is to look at the following class of pseudocharacters: However, it has to be discarded as the following observation shows.
Lemma 4.1 Let f ∈ P X (G) and suppose that (11) f holds for all x ∈ G and all b in a certain subset B ⊆ G. Then (11) also holds for all x ∈ G and all b in the normal subgroup N ⊆ G generated by B. Moreover, if f (x b ) = f (x) for all x ∈ G and b ∈ B, then the same is true for all x ∈ G and all b ∈ N .
Proof. It suffices to show that for all y ∈ G, i.e., b −1 ∈ H . Finally, for fixed b ∈ H , g ∈ G and an arbitrary x ∈ G we have proving that g b g −1 ∈ H , hence our first assertion. The argument for the second assertion is similar: one shows that is a normal subgroup of G.
Corollary 4.2 Suppose that G has the property that every nontrivial normal subgroup has finite index. If the center of G is trivial, then given a nonzero pseudocharacter f ∈ P X (G) and an element a ∈ G, a = e, there exists x ∈ G such that f (xa) = f (x) + f (a). Further analysis leads to the following definition.

Definition 4.3
Let H be a subgroup of a group G. We say that a quasicharacter We say that a pseudocharacter f ∈ P X (G) is H -spherical if there exists a strongly H -spherical quasicharacter F such that the difference f − F is bounded.

Example 4.4
We briefly recall the construction of quasicharacters used to prove the result in [7] (this construction was not provided explicitly in the original paper, and a similar construction with more geometric flavor was discovered independently in [5]) as these are essentially the only known quasicharacters on amalgamated products. A product u 1 · · · u n in the amalgamated product G = A 1 * H A 2 is called reduced if (i) every u i belongs to either A 1 or A 2 ; (ii) u i and u i+1 belong to different factors A j , j = 1, 2; (iii) if n > 1 then none of u i belongs to H ; (iv) if n = 1 then u 1 = 1. Grigorchuk's construction is based on the fact that if u 1 · · · u n = v 1 · · · v m are two reduced products in G then n = m and for every i = 1, . . . , n, the elements u i and v i belong to the same double coset modulo H , which is a simple consequence of the structure theorem for reduced words in amalgamated products. Two words u and v are called generally equal if there exist reduced products u = u 1 · · · u n and v = v 1 · · · v m such that n = m and for every i = 1, . . . , n, the elements u i and v i belong to the same double coset modulo H . A reduced word w = w 1 · · · w k is said to generally occur in a reduced word u = u 1 · · · u n if there is a subword u i · · · u i+k−1 of u which is generally equal to w 1 · · · w k . We define # w (u) as the number of general occurrences of w in u and for any g ∈ G we let where u is any reduced word representing g . It turns out that if w is a reduced word then the function F w is a quasicharacter of G. In case |H \A 1 /H | 3 and [A 2 : H ] 2 it is possible to exhibit an infinite sequence of reduced words {w n } such that the quasicharacters {F w n } are linearly independent, whence the infinite dimensionality of the second bounded cohomology group. It is immediate from (12) that Grigorchuk's quasicharacters are strongly H -spherical.
Notice that if F is a strongly H -spherical quasicharacter, then the restriction of F to H is a character of H , in particular, F (1) = 0. Also, if H 1 and H 2 are subgroups of a group G and F is a strongly H i -spherical quasicharacter for i = 1, 2, then F is a strongly H -spherical quasicharacter where H is the subgroup of G generated by H 1 and H 2 . We denote the space of H -spherical pseudocharacters of G by P X (G) H .
In the sequel we will need the following observation.
Proof. Our claim follows from gives rise to the following embedding of the spaces of pseudocharacters which allows us to identify the former with a subspace of the latter. We denote the kernel of the linear map by P X 0 (A 1 * A 2 ). The following analog of Theorem 2.1 holds for H -spherical pseudocharacters in the case when the amalgamated subgroup H is arbitrary.
Theorem 4.6 Let H be an arbitrary subgroup of A 1 and A 2 , θ: be the subgroup of A 1 * A 2 generated by H * H and Ker θ, and P X 0,Ker θ (A 1 * A 2 ) be the subspace of P X 0 (A 1 * A 2 ) consisting of pseudocharacters with trivial restriction to Ker θ. Then the sequence of vector spaces is exact.
5. PROOF OF THEOREM 4.6 5.1. Exactness in the term P X (A 1 ) H ⊕ P X (A 2 ) H . To prove the exactness of (13) in the term P X (A 1 ) H ⊕ P X (A 2 ) H we need to show that given H -spherical pseudochar- be strongly H -spherical quasicharacters with the property that the differences F i − f i are bounded; also let C = max{C F 1 , C F 2 }. An analog of Lemma 3.1 shows that for the existence of f it suffices to construct a strongly H -spherical quasicharacter F ∈ QX (A 1 * H A 2 ) with the property that the differences F | A i − F i are bounded for i = 1, 2.
Let X i be an arbitrary system of representatives of left cosets = H in A i /H , i = 1, 2, and let X = X 1 ∪ X 2 . Similarly to §3, we introduce a function F on X whose restriction to X i is F i : x ∈ X i , and let W be the set of all words of the form x 1 · · · x n where x i ∈ X and for every i = 1, . . . , n − 1, the elements x i and x i+1 belong to different parts X 1 or X 2 of X (by convention, the empty word is included in W and corresponds to n = 0). Then any element g ∈ G := A 1 * H A 2 admits a unique canonical presentation of the form (14) g = x 1 · · · x n h for some h ∈ H and some word x 1 · · · x n ∈ W .
Since the restrictions of f i to H coincide, the difference F 1 | H − F 2 | H is bounded. However, the restrictions F i | H , i = 1, 2, are the characters of H , hence and we let F 0 denote the common restriction of F 1 and F 2 to H : We now extend F to a function on A 1 * H A 2 using the canonical form (14): To complete the proof of exactness of (13) in P X (A 1 ) H ⊕ P X (A 2 ) H it suffices to establish the following. The property that the differences F | A i − F i are bounded for i = 1, 2, follows immediately from (15) (moreover, F | A i = F i ).
Next, we are going to show that F is a quasicharacter of A 1 * H A 2 . When we switch a representative of a coset modulo H and an element of H , both of them will change. Since it is necessary to keep track of all these changes, we introduce the following notation: given elements x ∈ X i and h ∈ H , there exist elements x 〈h〉 ∈ X i and h 〈x〉 ∈ H such that (16) h x = x 〈h〉 h 〈x〉 .
To simplify notation we will write h 〈x 1 ,x 2 〉 instead of h 〈x 1 〉 〈x 2 〉 and similarly for x 〈h 1 ,h 2 〉 . From (16) we derive that which is a crucial equality in our argument. One of the main consequences of this equality is the following fact which follows from (17) by induction on m. + · · · + F y 〈h 〈y 1 ,...,y m−1 〉 〉 m + F h 〈y 1 ,...,y m 〉 = F (y 1 ) + · · · + F (y m ) + F (h).
In the general case, there might be some cancelation in the middle in the product g 1 g 2 , and we indicate several steps to write the canonical form of g 1 g 2 in a convenient way. First, we write it in the form (which is not a canonical form in general) h 2 ∈ H , then (18) becomes g 1 g 2 = x 1 · · · x m z 1 · · · z n h 0 . It remains to consider the case when x m and z 1 belong to the same factor A i ; then where u 1 ∈ X or u 1 = e and a 1 ∈ H . If u 1 ∈ X then · · · z 〈a 〈z 2 ,...,z n−1 〉 1 〉 n a 〈z 2 ,...,z n 〉 1 h 0 is the canonical form of g 1 g 2 . If u 1 = e then g 1 g 2 = x 1 · · · x m−1 a 1 z 2 · · · z n h 0 = x 1 · · · x m−1 z 〈a 1 〉 2 a 〈z 2 〉 1 z 3 · · · z n h 0 . Notice that we do not transfer a 1 all the way to the right. Since x m−1 and z 2 must belong to the same X i , we next write where u 2 ∈ X or u 2 = e and a 2 ∈ H . We continue this process until we find a positive integer k such that k−1 = u k a k where u k ∈ X and a k ∈ H . Then the canonical form of g 1 g 2 is Before we can estimate |(δF )(g 1 , g 2 )| we need the following fact.
(Lemma 5.2) To finish the proof of Proposition 5.1 it remains to show that F satisfies properties (i) and (ii) of Definition 4.3.
To prove property (i) we write an arbitrary g ∈ G in the canonical form g = x 1 · · · x n h; then for any h 1 , h 2 ∈ H the canonical form of h 1 g h 2 is Since the restriction of F to H is a character of H we obtain and Lemma 5.2 implies that F (h 1 g h 2 ) = F (x 1 ) + F (x 2 ) + · · · + F (x n ) + F (h 1 ) + F (h) + F (h 2 ) = F (h 1 ) + F (g ) + F (h 2 ), as required.
To prove property (ii) we first suppose that the canonical form of g ∈ G is x 1 · · · x n , i.e., there is no H -component. Then the canonical form of g −1 is x −1 n · · · x −1 1 and (20) In the general case write g = g 0 h, where the canonical form of g 0 has no H -component.
Since we already showed that F satisfies property (i) of strongly H -spherical quasicharacters, we use (20) to obtain as required.

5.2.
Exactness in the term P X (A 1 * H A 2 ) H . Given f ∈ P X 0 (A 1 * H A 2 ) H = Ker β, we let F denote the corresponding strongly H -spherical quasicharacter of A 1 * H A 2 . We claim thatF := F • θ is a strongly -spherical quasicharacter of A 1 * A 2 . Indeed, the boundedness of δF follows from that of δF and, moreover,F is both strongly H * H -spherical (since θ(H * H ) = H ) and strongly Ker θ-spherical, hence is strongly -spherical. There exists a pseudocharacterf ∈ P X (A 1 * A 2 ) such that the differencẽ F −f is bounded. Thusf is an -spherical pseudocharacter of A 1 * A 2 and clearlỹ This shows the inclusion Ker β ⊆ P X 0,Ker θ (A 1 * A 2 ) .
For the opposite inclusion we considerf ∈ P X 0,Ker θ (A 1 * A 2 ) and letF be the corresponding strongly -spherical quasicharacter. Theñ F (g x) =F (g ) +F (x) for all g ∈ A 1 * A 2 and all x ∈ Ker θ.
Sincef | Ker θ = 0, the restrictionF | Ker θ is bounded. But Ker θ ⊆ andF is a character of . We conclude thatF | Ker θ = 0 and thus F (g x) =F (g ) for all g ∈ A 1 * A 2 and all x ∈ Ker θ. Therefore there exists a function F on A 1 * H A 2 such thatF = F • θ. It is immediate that F is a strongly H -spherical quasicharacter of A 1 * H A 2 with bounded restrictions to A 1 and A 2 . Hence we can construct an H -spherical pseudocharacter f of A 1 * H A 2 whose restrictions to A 1 and A 2 are trivial. Therefore Ker β ⊇ P X 0,Ker θ (A 1 * A 2 ) and the proof of Theorem 4.6 is complete.