ON BOUNDED COHOMOLOGY OF AMALGAMATED PRODUCTS OF GROUPS

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ON BOUNDED COHOMOLOGY OF AMALGAMATED PRODUCTS OF GROUPS

IGOR V. EROVENKO Received 12 November 2003

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.

2000 Mathematics Subject Classification: 20J06, 20E06.

1. Introduction. We recall that bounded cohomologyH_b^∗(G)of a groupG(we will be considering only cohomology with coefficients in the additive group of realsRwith triv- ial action, so, in our notations for cohomology, the coefficient module will be omitted) is defined using the complex

··· ← C_bⁿ⁺¹(G) ^δ

nb

←C_bⁿ(G)← ··· ← C_b²(G)←^δ1b C_b¹(G)←^δ R^0b=0 ←^{δ−1b =0} 0 (1.1)

of bounded cochainsf:G×···×G→R, andδⁿ_b=δⁿ|Cⁿ_b(G)is the bounded differential operator. SinceH_b⁰(G)=RandH_b¹(G)=0 for any groupG, investigation of bounded co- homology starts in dimension 2. One observes thatH_b²(G)contains a subspaceH²_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 quo- tient spaceH_b²(G)/H_b,2² (G)is canonically isomorphic to the bounded part of the ordi- nary cohomology groupH²(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 functionF:G→Ris called aquasicharacterif there exists a constant C_F0 such that

F(xy)−F(x)−F(y) C_F ∀x,y∈G. (1.2)

A functionf:G→Ris called apseudocharacteriffis a quasicharacter and, in addition,

f gⁿ

=nf (g) ∀g∈G, n∈Z. (1.3)

The notions of a quasicharacter and a pseudocharacter originally arose from the ques- tions of stability of solutions of functional equations [9,10,11] and continuous repre- sentations of groups [12]. We use the following notation:

(i) X(G)is the space of additive charactersG→R; (ii) QX(G)is the space of quasicharacters;

(iii) PX(G)is the space of pseudocharacters;

(iv) B(G)is the space of bounded functions.

Then

H_b,2² (G)QX(G)/X(G)⊕B(G)

PX(G)/X(G) (1.4) as vector spaces (cf. [6, Proposition 3.2 and Theorem 3.5]). Special interest inH_b,2² 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 triv- iality ofH_b,2² for Chevalley groups over rings ofS-integers in algebraic number fields using bounded generation.) For example, Grigorchuk [7] (cf. also [5]) proved that the amalgamated productA1∗HA2does not have bounded generation provided that the number of double cosets ofA1moduloHis at least 3 and[A2:H] 2 by showing that dimH_b,2² (A1∗HA2)= ∞in this case. The proof is based on the explicit construction (seeExample 4.4) of an infinite family of linearly independent quasicharacters which naturally generalize the construction of quasicharacters for free groups. The quasichar- acters for free groups were first constructed by Brooks [2], and Fa˘ıziev 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 productA1∗HA2is the Mayer-Vietoris exact sequence (see [14, Theorem 2.3])

··· →Hⁿ

A1∗HA2

→Hⁿ A1

⊕Hⁿ A2

→Hⁿ(H) →Hⁿ⁺¹

A1∗HA2

→ ···,

(1.5) 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 where 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 callH-spherical (Theorem 4.6); seeSection 4for relevant definitions and discussion.

The sequences (2.11) and (4.14) constructed in Theorems2.1and4.6, 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 spaces 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.

Lemma1.1. Any pseudocharacter is constant on conjugacy classes; a bounded pseu- docharacter is trivial.

Proof. Letf ∈PX(G) and suppose that f (yxy⁻¹)−f (x)=a =0 for some x, y∈G. Then the differencef (yxⁿy⁻¹)−f (xⁿ)=nais unbounded whenn→ ∞. On the other hand,

f

yxⁿy⁻¹

−f

xⁿ=f

yxⁿy⁻¹

−f (y)−f xⁿ

−f

y⁻¹2C_f, (1.6) a contradiction. The second assertion is obvious.

2. 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 assum- ing that the amalgamated subgroupNis normal in both factorsA1andA2(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

β:PX

A1∗NA2

→PX A1

⊕PX A2

(2.1)

asβ=(β1,β2), where

βi:PX

A1∗NA2

→PX Ai

, i=1,2, (2.2)

is the restriction map associated with the natural embeddingAiA1∗NA2. Next, let γ:PX

A1

⊕PX A2

→PX(N) (2.3)

be defined by

γ f1,f2

=f1_N−f2_N. (2.4)

In contrast to the usual Mayer-Vietoris sequence for the spaces of characters, 0 →XA1∗NA2 β^˜

→XA1

⊕XA2 ˜γ

→X(N), (2.5)

where ˜β and ˜γ are analogous to βand γ introduced above, the sequence for pseu- docharacters 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:PX A1/N

∗ A2/N

→PX

A1∗NA2

(2.6)

induced by the natural surjective homomorphism A1∗NA2 →

A1/N

∗ A2/N

(2.7)

and letPX0((A1/N)∗(A2/N))denote the kernel of the linear map β¯:PXA1/N

∗A2/N

→PXA1/N

⊕PXA2/N, (2.8) β¯=(β¯1,β¯2), where

β¯_i:PXA1/N

∗A2/N

→PXA_i/N

(2.9) is the restriction map induced by the natural embedding

Ai/N A1/N

∗ A2/N

, i=1,2. (2.10)

The above spaces and linear maps align in the following sequence.

Theorem2.1. LetNbe a normal subgroup ofA1andA2. Then the sequence of vector spaces

0→PX0A1/N

∗A2/N α

→PXA1∗NA2 β→PX

A1

⊕PX A2

γ

→PX(N) (2.11)

is exact.

We will prove the theorem in the next section and will now derive two consequences.

Corollary2.2. Given two arbitrary pseudocharactersf1andf2on the groupsA1

andA2, respectively, there exists a pseudocharacterf on the free productA1∗A2such thatf|A_i=fi,i=1,2.

Corollary2.3. LetAbe an arbitrary group andNits normal subgroup. Then the re- striction homomorphismρ:PX(A∗NA)→PX(A), induced by embeddingAintoA∗N A as either factor, is surjective. If, moreover,[A:N]=2, thenρis an isomorphism.

Proof. For the first assertion, one needs to observe that for anyf∈PX(A), the pair (f ,f )belongs to Kerγ, and is therefore obtained as the restriction of a pseudocharacter onA∗NA. If[A:N]=2, then(A/N)∗(A/N)is the infinite dihedral group. Since it is amenable, all its pseudocharacters are in fact characters [6, Theorem 2.1]. On the other hand, since it is generated by elements of order two, it does not have nonzero characters. This, in particular, implies that

PX0

A1/N

∗ A2/N

=0, (2.12)

hence our second claim.

3. Proof ofTheorem 2.1

3.1. Exactness in the termPX(A1)⊕PX(A2). The inclusion Imβ⊂Kerγbeing obvi- ous, all we need to prove is that, given pseudocharactersfi∈PX(Ai),i=1,2, satisfying f1|N=f2|N, there exists a pseudocharacterf∈PX(A1∗NA2)such thatf|A_i=fi. The following observation saves the (serious) trouble of verifying the conditionf (gⁿ)= nf (g).

Lemma3.1. In the current notation, for the existence of a pseudocharacterf, it suf- fices to construct aquasicharacterF∈QX(A1∗NA2)such that the differencesF|A_i−fi

are bounded fori=1,2.

Proof. Indeed, it follows from (1.4) that given such anF, there exists a pseudochar- acterf∈PX(A1∗NA2)for which the differenceF−f is bounded. Then, fori=1,2, the difference

fi−f|A_i=

fi−F|A_i

+(F−f )_A

i (3.1)

is a bounded pseudocharacter ofA_i, hence zero, proving thatf|Ai=f_i.

The construction of such a quasicharacter F ∈ QX(A1∗NA2) 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 representativesX_ihaving the follow- ing property:

(P1) ifx,y∈X_iandxy∈N, then eitherx²,y²∈Nory=x⁻¹.

Indeed, let ¯S_i denote the set of elements of order two inA_i/N and pick an arbitrary system of representativesSi⊂Aiof the cosets from ¯Si. Since, for each

x∈T¯_i:=A_i/N

\S¯_i∪{e}

(3.2) (ethe identity), we havex =x⁻¹, there exists a partition ¯Ti=T¯_i∪T¯_isuch that ¯T_i∩T¯_i=

∅and ¯T_i=(T¯_i)⁻¹. Choose an arbitrary system of representativesT_i⊂Aiof the cosets from ¯T_iand letT_i=T_i∪(T_i)⁻¹. Finally, letX_i=S_i∪T_i.

Suppose that the systems of representativesXiwith property (P1) have been chosen.

We define an involutive transformationτi:Xi→Xiby setting τ_i(x)=





x ifx∈S_i,

x⁻¹ ifx∈Ti. (3.3)

Now, we letX=X1∪X2(disjoint union) and introduce a functionF and an involution τonXwhose restrictions toXiarefiandτi, respectively:

F(x)=f_i(x), τ(x)=τ_i(x) ifx∈X_i. (3.4) Let W be the set of all words of the form x1···xn, where xi ∈ X and, for every i=1,...,n−1, the elementsx_i and x_i+1belong todifferent partsX1 orX2ofX (by convention, the empty word is included inW and corresponds ton=0). Then each elementg∈G:=A1∗NA2admits a unique canonical presentation of the form

g=x1···x_nh (3.5)

for some h∈Nand some word x1···xn∈W (cf., e.g., [13, Chapter I, Theorem 1]).

Using the canonical form (3.5), we can extendτ to an involutive transformation ofG by setting

τ(g)=τ xn

···τ x1

h. (3.6)

Letf0∈PX(N)denote the common restriction off1andf2toN:

f0:=f1|N=f2|N. (3.7)

It follows from (3.6) that for every g∈ G, we have gτ(g)∈ N, so the expression f0(gτ(g)) makes sense. LetS =S1∪S2 and T =T1∪T2. We extendF to a function onA1∗NA2by the formula

F(g)=µ(g)+η(g), (3.8)

where

µ(g)=1 2f0

gτ(g)

, η(g)=

x_i∈T

F xi

, (3.9)

and, by convention,η(g)=0 ifg∈N(in this definition, we use the unique canonical presentation (3.5)).

Proposition 3.2. The function F defined by (3.8) is a quasicharacter ofA1∗NA2

such that the differencesF|Ai−f_iare bounded fori=1,2.

First, we observe that forh∈N, we have µ(h)=1

2f0h²

=f0(h). (3.10)

In particular,F|N=µ|Nis a pseudocharacter with constantC=Cf₀. Also, writingg∈G in the canonical form and using the fact thatf0 is the common restriction of pseu- docharactersf1andf2toN, we obtain

µ ghg⁻¹

=µ(h) ∀g∈G, h∈N. (3.11)

Next, we will show that the differenceF|Ai−f_iis a bounded function onA_ifori=1,2.

Forg∈N, we have

F(g)=1 2f0

g²

=f0(g). (3.12)

Ifg∈A_i\N, we write it in the formg=xhwithh∈N,x∈X_i. Ifx∈S_i, then F(g)−f_i(g)=

1

2f0(xhxh)−f_i(xh)

= 1

2f_ix²x⁻¹hx h

−f_i(xh) 1

2fi x²

x⁻¹hx h

−fi x²

−2fi(h) +f_i(x)+f_i(h)−f_i(xh)

2Cf_i.

(3.13)

Ifx∈T_i, then

F(g)−f_i(g)= 1

2f0xhx⁻¹h

+f_i(x)−f_i(xh) 1

2fi xhx⁻¹

h

−2fi(h)+fi(h)+fi(x)−fi(xh) 3

2Cf_i.

(3.14)

In particular, we obtain

F|A_i∈QX Ai

. (3.15)

To complete the proof of the proposition, it remains to show thatFis a quasicharacter on the entire amalgamated productA1∗NA2. For a functionf onG, we define

(δf ) g1,g2

=f g1g2

−f g1

−f g2

. (3.16)

So, we need to show thatδF=δµ+δηis bounded onG×G.For convenience of further reference, we will collect, in the following lemma, some properties of the functionsµ andη.

Lemma3.3. (i)Ifg∈G,h∈N, then|µ(gh)−µ(g)−µ(h)| C.

(ii) Ifg1,...,gk∈Gandτ(g1···gk)=τ(gk)···τ(g1), then

µ

g1···g_k

− k i=1

µ g_i

k−1

2 C. (3.17)

(iii) η(gh)=η(g)for anyg∈Gand anyh∈N.

(iv) If x1···x_n∈ W, then η(x1···x_n)=η(x1···x_i)+η(x_i+1···x_n) for any i= 1,...,n−1.

(v) η(τ(g))= −η(g)for anyg∈G.

Proof. Indeed, ifg∈Gandh∈N, thenµ(gh)=(1/2)f₀(ghτ(g)h), whence µ(gh)−µ(g)−µ(h)=1

2f0gτ(g)τ(g)⁻¹hτ(g)h

−f0gτ(g)

−f0h², (3.18) and (i) follows. For (ii), one needs to observe that

µ

g1···gk

=1 2f0

g1···g_k−1 gkτ

gk

g1···g_k−1−1

×

g1···g_k−2g_k−1τg_k−1g1···g_k−2−1

···g1τg1 . (3.19) Properties (iii), (iv), and (v) follow immediately from the definition ofη.

For two given elementsg1,g2∈G, we pick the canonical presentations

g1=x1···xmh1, g2=y1···ynh2, (3.20)

wherex1···x_m,y1···y_n∈Wandh1,h2∈N. We first consider the easiest case where xmandy1belong to different factorsAi,i=1,2. In this case, the canonical presentation ofg1g2is

g1g2=x1···x_my1···y_nh, (3.21) where

h=

y1···y_n−1h1y1···y_nh2∈N. (3.22) It follows from Lemma 3.3(iii) and (iv) that (δη)(g1,g2) = 0, so (δF)(g1,g2) = (δµ)(g1,g2). Since

τx1···x_my1···y_n

=τy1···y_nτx1···x_m, (3.23) we conclude fromLemma 3.3(i) and (ii) and (3.11) that

µg1g2

−µh1

−µh2

−µx1···x_m

−µy1···y_n µ

x1···x_my1···y_nh

−µ

x1···x_my1···y_n

−µ(h) +µ(h)−µ

h1

−µ h2 +µ

x1···xmy1···yn

−µ

x1···xm

−µ

y1···yn C+C+C

2 =5 2C.

(3.24)

On the other hand, µ

g1

+µ g2 −

µ h1

+µ h2

+µ

x1···xm +µ

y1···yn 2C. (3.25) It follows, in this case, that

(δF)

g1,g29

2C. (3.26)

To consider the general case, we need to introduce the fragments ofg1andg2that cancel out ing1g2. Letkbe the largest integer less than or equal to min{m,n}such thatx_m−i+1yi∈Nfor alli=1,...,k. We introduce the following elements:

w1=x1···x_m−k−1, u1=x_m−k, v1=x_m−k+1···xm,

v2=y1···yk, u2=y_k+1, w2=y_k+2···yn, (3.27) where, by convention,v1=eifk=0,w1=eifm=k+1, andw1=u1=eifm=k, with similar rules forv2,u2, andw2. We observe thatv2=τ(v1), so, lettingv=v1, we have the following factorizations:

g1=w1u1vh1, g2=τ(v)u2w2h2. (3.28) It follows from our construction that bothu1andu2belong to thesamefactorA_i, so we can write

u1u2=zh for someh∈N, z∈X. (3.29)

We claim that

(δµ) g1,g2

−(δµ)

u1,u210C. (3.30)

This estimation is a consequence of the following three inequalities that reflect the three-step transition fromg₁,g₂tou₁,u₂.

Lemma3.4. (i)Lets₁=w₁u₁v,s₂=τ(v)u₂w₂, so thatg_i=s_ih_i,i=1,2; then (δµ)g1,g2

−(δµ)s1,s24C. (3.31)

(ii) Lett1=w1u1,t2=u2w2, so thats1=t1v,s2=τ(v)t2; then (δµ)

s1,s2

−(δµ)

t1,t22C. (3.32)

(iii) |(δµ)(t1,t2)−(δµ)(u1,u2)|4C.

Proof. (i) We have

g1g2=s1s2s⁻₂¹h1s2 h2. (3.33) UsingLemma 3.3(i) and (3.11), we obtain the following two inequalities:

µ g1g2

−µ s1s2

−µ h1

−µ

h22C, µ

g1

+µ g2

− µ

s1

+µ s2

+µ h1

+µ

h22C, (3.34)

from which (i) follows.

(ii) We have

s1s2=t1t2

t₂⁻¹

vτ(v) t2

. (3.35)

UsingLemma 3.3(i) and (3.11), we obtain µ

s1s2

−µ t1t2

−µ

vτ(v) C. (3.36)

Sinceτ(s1)=τ(v)τ(t1)andτ(s2)=τ(t2)v,Lemma 3.3(ii) combined with the observa- tion thatµ(τ(v))=µ(v)implies that

µ s1

+µ s2

− µ

t1

+µ t2

+2µ(v) C. (3.37)

Butµ(vτ(v))=f0(vτ(v))=2µ(v), and (ii) follows.

(iii) We have

t1t2=w1zw2

w₂⁻¹hw2

. (3.38)

Sinceτ(w1zw2)=τ(w2)τ(z)τ(w1),Lemma 3.3(i) and (ii) and (3.11) imply that µ

t1t2

− µ

w1

+µ(z)+µ w2

+µ(h)2C, (3.39)

and therefore

µt1t2

−µw1

+µzh

+µw23C. (3.40)

On the other hand, sinceτ(t1)=τ(u1)τ(w1)andτ(t2)=τ(w2)τ(u2), we have µ

t1

+µ t2

− µ

w1

+µ u1

+µ w2

+µ

u2 C. (3.41)

Sincezh=u1u2, we obtain (iii).

Next, we calculate(δη)(g1,g2)usingLemma 3.3(iii), (iv), and (v):

(δη) g1,g2

= η

w1

+η(z)+η w2 −

η w1

+η u1

+η(v)

−ητ(v)

+ηu2

+ηw2 =η(z)−ηu1

−ηu2

=(δη)u1,u2.

(3.42)

From (3.30) and (3.42), we obtain (δF)

g1,g2

−(δF)

u1,u210C. (3.43)

On the other hand, since bothu1andu2belong to the same factorAi, (3.15) implies that (δF)(u1,u2)is bounded, and we finally conclude that (δF)(g1,g2)is bounded, completing the proof ofProposition 3.2.

3.2. Exactness in the termPX(A1∗NA2). The exactness of (2.11) inPX(A1∗NA2) is based on the following fact.

Lemma3.5. LetGbe an arbitrary group andNits normal subgroup. If a pseudochar- acterf∈PX(G)has zero restriction toN, then it satisfiesf (gh)=f (g)for allh∈N, g∈G. In other words, the natural sequence

0 →PX(G/N)→PX(G) →PX(N) (3.44) is exact.

Proof. Suppose thatf (gh) =f (g)for someh∈N,g∈G, and leta=f (gh)−f (g).

Then

f (gh)ⁿ

−f

gⁿ=n|a| → ∞ whenn → ∞. (3.45) On the other hand,(gh)ⁿcan be written asgⁿhforh∈N, so

f (gh)ⁿ

−f

gⁿ=f gⁿh

−f gⁿ

−f

h (3.46)

is bounded independent ofn.

Iff∈Kerβ, thenf|N=0.Lemma 3.5implies thatf factors through the group ho- momorphism

A1∗NA2 →

A1∗NA2

/N A1/N

∗ A2/N

, (3.47)

immediately implying thatf∈Imαand proving the inclusion Kerβ⊂Imα. The oppo- site inclusion is obvious.

3.3. Remarks. The general construction of a quasicharacterF∈QX(A1∗NA2)lift- ing given pseudocharactersfi∈PX(Ai) (i=1,2)with the same restrictions toNes- sentially simplifies in the following two particular cases:

(1) S= ∅, that is, when the quotientsA1/NandA2/Ndo not have elements of order two;

(2) T= ∅, that is, 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”:

F

x1···xnh

= n i=1

F xi

+f0(h). (3.48)

4. The case of an arbitrary subgroup. If we do not assume that the amalgamated subgroupHis normal in both factorsA1andA2, then two difficulties arise. First of all, when we switch representatives of cosets moduloHwith elements ofHin 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 stronglyH-sphericalandH-spherical, respectively. We would like to point out that the only explicitly known quasicharacters on amalgamated products (seeExample 4.4) are stronglyH-spherical. Below is a brief analysis of what restrictions should be imposed on pseudocharacters.

LetHbe a subgroup of a groupG. The first conjecture, which naturally arises after preliminary considerations, is to look at the following class of pseudocharacters:

f∈PX(G)|f (xh)=f (x)+f (h)∀x∈G, h∈H. (4.1)

However, it has to be discarded as the following observation shows.

Lemma4.1. Letf∈PX(G)and suppose that

f (xb)=f (x)+f (b) (4.2) holds for allx∈Gand allbin a certain subsetB⊂G. Then (4.2) also holds for allx∈G and allbin the normal subgroupN⊂Ggenerated byB. Moreover, iff (xb)=f (x)for allx∈Gandb∈B, then the same is true for allx∈Gand allb∈N.

Proof. It suffices to show that H=

b∈G|f (xb)=f (x)+f (b)∀x∈G

(4.3) is a normal subgroup ofG. Ifb1,b2∈H, then

f x

b1b2

=f xb1

b2

=f (x)+f b1

+f b2

=f (x)+f b1b2

(4.4)

for allx∈G, sob1b2∈H. Similarly, forb∈H, we have f (x)=f

(xb)b⁻¹

=f (xb)−f (b)=f (xb)+f b⁻¹

(4.5) which means that

f yb⁻¹

=f (y)+f b⁻¹

(4.6) for ally∈G, that is,b⁻¹∈H. Finally, for fixedb∈H,g∈G, and an arbitraryx∈G, we have

fxgbg⁻¹

=fg⁻¹xgb

=fg⁻¹xg

+f (b)=f (x)+fgbg⁻¹, (4.7) proving thatgbg⁻¹∈H, hence our first assertion. The argument for the second asser- tion is similar: one shows that

b∈G|f (xb)=f (x)∀x∈G

(4.8) is a normal subgroup ofG.

Corollary4.2. Suppose thatGhas the property that every nontrivial normal sub- group has finite index. If the center ofGis trivial, then, given a nonzero pseudochar- acterf ∈PX(G)and an element a∈G,a =e, there exists x∈Gsuch that f (xa) = f (x)+f (a).

Proof. Assume the contrary. Then, according toLemma 4.1, the equalityf (xb)= f (x)+f (b)holds for allx∈Gand allbinN:=the normal subgroup ofGgenerated bya. In particular, the restrictionf|Nis a character ofN. Moreover, sincefis constant on conjugacy classes inG, it vanishes on the commutator subgroup[G,N]. Since the center ofGis trivial,[G,N] = {e}and therefore has finite index inG.But thenf|[G,N]=0 impliesf=0 as required.

Further analysis leads to the following definition.

Definition4.3. LetHbe a subgroup of a groupG. A quasicharacterF∈QX(G)is stronglyH-sphericalif

(i) F(h1gh2)=F(h1)+F(g)+F(h2)for allh1,h2∈Handg∈G;

(ii) F(g⁻¹)= −F(g)for allg∈G.

A pseudocharacterf∈PX(G)isH-sphericalif there exists a stronglyH-spherical qua- sicharacterF such that the differencef−F is bounded.

Example4.4. We briefly recall the construction of quasicharacters used to prove the result in [7] (a similar construction with a more geometric flavor was discovered independently in [5]) as these are essentially the only known quasicharacters on amal- gamated products. A productu1···un in the amalgamated productG=A1∗HA2 is calledreducedif the following holds:

(i) everyu_ibelongs to eitherA1orA2;

(ii) uiandu_i+1belong to different factorsAj,j=1,2;

(iii) ifn >1, then none ofu_ibelongs toH;

(iv) ifn=1, thenu1 =1.

Grigorchuk’s construction is based on the fact that if u1···un =v1···vm are two reduced products inG, thenn=m and, for everyi=1,...,n, the elementsu_i and vibelong to the same double coset moduloH, which is a simple consequence of the structure theorem for reduced words in amalgamated products. Two wordsuandvare called generally equal if there exist reduced productsu=u1···u_nandv=v1···v_m such thatn=mand, for everyi=1,...,n, the elementsu_iandv_ibelong to the same double coset moduloH. A reduced wordw=w1···wkis said togenerally occur in a reduced wordu=u1···u_nif there is a subwordu_i···u_i+k−1ofuwhich is generally equal tow1···w_k. We define #_w(u)as the number of general occurrences ofwinu, and, for anyg∈G, we let

F_w(g)=#_w(u)−#_w−1(u), (4.9) whereuis any reduced word representingg. It turns out that ifwis a reduced word, then the functionFw is a quasicharacter ofG. In case|H\A1/H|3 and[A2:H] 2, it is possible to exhibit an infinite sequence of reduced words{w_n}such that the qua- sicharacters{Fwn}are linearly independent, whence the infinite dimensionality of the second bounded cohomology group. It is immediate from (4.9) that Grigorchuk’s qua- sicharacters are stronglyH-spherical.

Notice that ifF is a stronglyH-spherical quasicharacter, then the restriction ofF to H is a character ofH; in particular,F(1)=0. Also, ifH1 andH2are subgroups of a groupGandF is a stronglyHi-spherical quasicharacter fori=1,2, thenFis a strongly H-spherical quasicharacter, whereHis the subgroup ofGgenerated byH1andH2. We denote the space ofH-spherical pseudocharacters ofGbyPX(G)_H.

In the sequel, we will need the following observation.

Lemma 4.5. If F is a strongly H-spherical quasicharacter andg1g2∈H for some g1,g2∈G, thenF(g1g2)=F(g1)+F(g2).

Proof. Our claim follows from F

g2

=F g₁⁻¹

+F g1g2

= −F g1

+F g1g2

. (4.10)

The canonical (surjective) homomorphism

θ:A1∗A2 →A1∗HA2 (4.11)

gives rise to the following embedding of the spaces of pseudocharacters:

ι:PX

A1∗HA2

PX A1∗A2

, (4.12)

which allows us to identify the former with a subspace of the latter. We denote the kernel of the linear map

β:PX A1∗A2

→PX A1

⊕PX A2

(4.13)

by PX0(A1∗A2). The following analog of Theorem 2.1 holds for H-spherical pseu- docharacters in the case when the amalgamated subgroupHis arbitrary.

Theorem4.6. LetHbe an arbitrary subgroup ofA1andA2, letθ:A1∗A2→A1∗HA2

be the canonical homomorphism, letᏴbe the subgroup ofA1∗A2generated byH∗H andKerθ, and letPX_0,Kerθ(A1∗A2)_Ᏼ be the subspace ofPX0(A1∗A2)_Ᏼ consisting of pseudocharacters with trivial restriction toKerθ. Then the sequence of vector spaces

0 →PX_0,KerθA1∗A2

Ᏼ →PXA1∗HA2

H

β→PX A1

H⊕PX A2

H

γ→PX(H) (4.14)

is exact.

5. Proof ofTheorem 4.6

5.1. Exactness in the termPX(A1)H⊕PX(A2)H. To prove the exactness of (4.14) in the termPX(A1)_H⊕PX(A2)_H, we need to show that givenH-spherical pseudocharac- tersf_i∈PX(A_i)_H, i=1,2, satisfying f1|H=f2|H, there exists an H-spherical pseu- docharacter f ∈ PX(A1∗HA2)H such that f|A_i = fi. Let Fi ∈QX(Ai), i= 1,2, be strongly H-spherical quasicharacters with the property that the differences F_i−f_i are bounded; also, let C = max{C_F1,C_F2}. An analog of Lemma 3.1 shows that for the existence off, it suffices to construct a stronglyH-spherical quasicharacterF ∈ QX(A1∗HA2)with the property that the differencesF|A_i−F_iare bounded fori=1,2.

LetX_ibe an arbitrary system of representatives of left cosets =HinA_i/H,i=1,2, and let X =X1∪X2. Similarly to Section 3, we introduce a function F on X whose restriction toX_iisF_i:

F(x)=F_i(x) ifx∈X_i, (5.1)

and letW be the set of all words of the form x1···xn, where xi∈X and for every i=1,...,n−1, the elementsxi and x_i+1belong to different partsX1 orX2ofX (by convention, the empty word is included inW and corresponds to n=0). Then any elementg∈G:=A1∗HA2admits a unique canonical presentation of the form

g=x1···xnh (5.2)

for someh∈Hand some wordx1···xn∈W.

Since the restrictions of f_i to H coincide, the difference F1|H−F2|H is bounded.

However, the restrictionsFi|H,i=1,2, are the characters ofH, hence

F1|H−F2|H=0 (5.3)

and we letF0denote the common restriction ofF1andF2toH:

F0:=F1|H=F2|H∈X(H). (5.4) We now extendF to a function onA1∗HA2using the canonical form (5.2):

F(g)=F x1

+···+F xn

+F0(h). (5.5)

To complete the proof of exactness of (4.14) inPX(A1)_H⊕PX(A2)_H, it suffices to es- tablish the following.

Proposition5.1. The functionFdefined by (5.5) is a stronglyH-spherical quasichar- acter ofA1∗HA2such that the differencesF|A_i−Fiare bounded fori=1,2.

The property that the differencesF|A_i−Fiare bounded fori=1,2 follows immedi- ately from (5.5) (moreover,F|Ai=F_i).

Next, we are going to show thatF is a quasicharacter ofA1∗HA2. When we switch a representative of a coset moduloHand an element ofH, both of them will change. Since it is necessary to keep track of all these changes, we introduce the following notation:

given elementsx∈X_iandh∈H, there exist elementsx^h∈X_iandh^x∈Hsuch that

hx=x^hh^x. (5.6)

To simplify notation, we will write h^x1,x2 instead of (h^x1)^x2 and similarly for x^h1,h2. From (5.6), we derive that

F(x)+F(h)=F x^h

+F h^x

(5.7) which is a crucial equality in our argument. One of the main consequences of this equality is the following fact which follows from (5.7) by induction onm.

Lemma5.2. Lety1,...,y_m∈Xandh∈H. Then F

y₁^h +F

y₂^hy1 +F

y₃^hy1,y2

+···+F

y_m^{hy1,...,ym−1} +F

h^y1,...,ym

=F y1

+···+F ym

+F(h). (5.8)

Given two elementsg1,g2∈G, we fix their canonical presentations

g1=x1···xmh1, g2=y1···ynh2, (5.9) wherex1···xm,y1···yn∈Wandh1,h2∈H. Suppose first thatxmandy1belong to different factorsAi,i=1,2. Then the canonical presentation ofg1g2is

g1g2=x1···x_my₁^h1y₂^hy11y^hy1

,y2

1

3 ···y_n^{hy11,...,yn−1}

h^y₁ ^1,...,ynh2

(5.10)

and (δF)

g1,g2

F y₁^h1

+F

y₂^hy11

+···+F

yn^{hy11,...,yn−1}

+F

h^y₁^1,...,yn

−Fy1

+···+Fy_n +Fh1

+C.

(5.11) Lemma 5.2implies that, in this case,|(δF)(g1,g2)| C.

In the general case, there might be some cancelation in the middle in the product g1g2, and we indicate several steps to write the canonical form ofg1g2in a convenient

way. First, we write it in the form (which is not a canonical form in general) g1g2=x1···xmy₁^h1y₂^hy11y^hy1

,y2

1

3 ···yn^{hy11,...,yn−1}

h^y₁^1,...,ynh2

(5.12)

and let

z1=y₁^h1, z2=y₂^hy11,..., z_n=yn^{hy11,...,yn−1}∈X, h0=h^y₁ ^1,...,ynh2∈H,

(5.13)

then (5.12) becomes

g1g2=x1···x_mz1···z_nh0. (5.14) It remains to consider the case whenx_mandz1belong to the same factorA_i; then

x_mz1=u1a1, (5.15)

whereu1∈Xoru1=eanda1∈H. Ifu1∈X, then g1g2=x1···x_m−1u1z^a₂¹z^a

z2

1

3 z^a

z2,z3

1

4 ···z^a

z2,...,zn−1

1

n

a^z₁^2,...,znh0

(5.16) is the canonical form ofg1g2. Ifu1=e, then

g1g2=x1···x_m−1a1z2···znh0=x1···x_m−1z^a₂¹a^z₁²z3···znh0. (5.17) Notice that we do not transfera1all the way to the right. Sincex_m−1andz2must belong to the sameX_i, we next write

x_m−1z^a₂¹a^z₁²=u2a2, (5.18) whereu2∈Xoru2=eanda2∈H. We continue this process until we find a positive integerksuch that

x_m−j+1z^a_j ^j−1a^z_j−1^j=aj∈H for 2 j k−1, (5.19) but

x_m−k+1z^a_k^k−1a^z_k−1^k=ukak, (5.20) whereuk∈Xandak∈H. Then the canonical form ofg1g2is

g1g2=x1···x_m−kukz^a_k+^k₁z^a

zk+1

k

k+2 ···z^a

zk+1,...,zn−1

k

n

a^z_k^k+1,...,znh0

. (5.21)

Before we can estimate|(δF)(g1,g2)|, we need the following fact.

Lemma5.3. In the current notation, F

uk +F

ak

− F

x_m−k+1

+···+F xm

+F z1

+···+F

zk C. (5.22)