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Low-dimensional bounded cohomology and extensions of groups

Bounded cohomology of groups was first studied by Gromov in 1982. Since then it has sparked much research in Geometric Group Theory. However, it is notoriously hard to explicitly compute bounded cohomology, even for most basic `non-positively curved' groups. On the other hand, there is a well-known interpretation of ordinary group cohomology in dimension 2 and 3 in terms of group extensions. The aim of this paper is to make this interpretation available for bounded group cohomology. This will involve quasihomomorphisms as defined and studied by Fujiwara and Kapovich.


Introduction
In [9] Gromov studied bounded cohomology of groups in connection to minimal volume of manifolds. Since then bounded cohomology has been established as an independent active research field due to its connection to other areas in Geometric Group Theory. Most prominent applications include stable commutator length ( [3]), circle actions ( [7], [8], [2]) and the Chern Conjecture. See [15] and [4] for an introduction to the topic.
For a group G and a normed G-module V , denote by H n b (G, V ) the n-dimensional bounded cohomology of G with coefficients in V ; see Subsection 2.2. H n b (G, V ) is notoriously hard to compute explicitly. Consider the most most basic case of V = R with a trivial G action. If G is amenable then it is known that H n b (G, R) = 0 for all n ≥ 1. On the other hand if G is "non-positively curved" then H 2 b (G, R) and H 3 b (G, R) are typically infinite dimensional as an R-vectorspace, for example for acylindrically hyperbolic groups; see [11] and [5]. However, there is no full characterisation of all bounded classes in H n b (G, R) for n = 2, 3. For n ≥ 4, H n b (G, R) is usually fully unknown, even if G is a non-abelian free group.
On the other hand, for ordinary n-dimensional group cohomology H n (G, V ) there is a wellknown characterisation for n = 2, 3 in terms of group extensions. The aim of this paper is to make this well-known correspondence available for bounded cohomology. For this, we first recall the classical connection between group extensions and ordinary group cohomology. Definition 1.1. An extension of a group G by a group N is a short exact sequence of groups We say that two group extensions 1 → N ι1 → E 1 π1 → G → 1 and 1 → N ι2 → E 2 π2 → G → 1 of G by N are equivalent, if there is an isomorphism Φ : E 1 → E 2 such that the diagram commutes.
Any group extension of G by N induces a homomorphism ψ : G → Out(N ); see Subsection 3.1. Two equivalent extensions of G by N induce the same such map ψ : G → Out(N ). We denote by E(G, N, ψ) the set of group extensions of G by N which induce ψ under this equivalence. If there is no danger of ambiguity we do not label the maps of the short exact sequence i.e. we will write 1 → N → E → G → 1 instead of (1).
It is well-known that one may fully characterise E(G, N, ψ) in terms of ordinary group cohomology: Theorem A. Let G and N be groups and let ψ : G → Out(N ) be a homomorphism. Furthermore, let Z = Z(N ) be the centre of N equipped with the action of G induced by ψ. Then there is a class ω = ω(G, N, ψ) ∈ H 3 (G, Z), called obstruction, such that ω = 0 in H 3 (G, Z) if and only if E(G, N, ψ) = ∅. In this case there is a bijection between the sets H 2 (G, Z) and E(G, N, ψ).
Theorem A may be found in Theorem 6.6 of [1], see also [12]. Moreover, for a G-module Z it is possible to characterise H 3 (G, Z) in terms of these obstructions: Theorem B. For any G-module Z and any α ∈ H 3 (G, Z) there is a group N with Z = Z(N ) and a homomorphism ψ : G → Out(N ) extending the action of G on Z such that α = ω(G, N, ψ).
Theorem B may be found in [1], Section IV, 6. In other words, any three dimensional class in ordinary cohomology arises as an obstruction.
The aim of this paper is to derive analogous statements to Theorem A and Theorem B involving bounded cohomology. This will use quasihomomorphisms as defined and studied by Fujiwara-Kapovich in [6]. Let G and H be groups. A set-theoretic function σ : G → H is called quasihomomorphism if the set D(σ) = {σ(g)σ(h)σ(gh) −1 |g, h ∈ G} is finite. We note that this is not the original definition of [6] but both definitions are equivalent; see Proposition 2.3 and Subsection 2.3. Definition 1.2. We say that an extension 1 → N ι → E π → G → 1 of G by N is bounded, if there is a (set theoretic) section σ : G → E such that (i) σ : G → E is a quasihomomorphism and (ii) the map φ σ : G → Aut(N ) induced by σ has finite image in Aut(N ).
We stress that φ σ is in general not a homomorphism. See Remark 2.4 for the notation. Condition (ii) may seem artificial but is both natural and necessary; see Remark 2.6. We denote the set of all bounded extensions of a group G by N which induce ψ by E b (G, N, ψ) and mention that this is a subset of E(G, N, ψ). 2 Analogously to Theorem A we will characterise the set E b (G, N, ψ) ⊂ E(G, N, ψ) using bounded cohomology.
Theorem C. Let G and N be groups and suppose that Z = Z(N ), the centre of N , is equipped with a norm · such that (Z, · ) has finite balls. Furthermore, let ψ : G → Out(N ) be a homomorphism with finite image.
There is a class then the bijection between the sets H 2 (G, Z) and E(G, N, ψ) described in Theorem A restricts to a bijection between im(c 2 ) ⊂ H 2 (G, Z) and E b (G, N, ψ) ⊂ E(G, N, ψ).
Here, c n : H n b (G, Z) → H n (G, Z) denotes the comparison map; see Subsection 2.2. We say that a normed group or module (Z, · ) has finite balls if for every K > 0 the set {z ∈ Z | z ≤ K} is finite. Theorem C is applied to examples in Subsection 5.1.
Just as in Theorem B we may ask which elements of H 3 b (G, Z) may be realised by obstructions. For a G-module Z we define the following subset of H 3 b (G, Z): Analogously to Theorem B we will show: Theorem D. Let G be a group, let Z be a normed G-module with finite balls and such that G acts on Z via finitely many automorphisms. Then and ψ induces the action on G} = F (G, Z) as subsets of H 3 b (G, Z). As finite groups are amenable this shows that all such classes in H 3 b (G, Z) will vanish under a change to real coefficients; see Subsection 2.2. We prove Theorem C and D following the outline of the classical proofs in [1].
1.1. Organisation of the paper This paper is organised as follows: In Section 2 we recall well-known facts of (bounded) cohomology and quasihomomorphisms. In Section 3 we will reformulate the problem of characterising group extensions using non-abelian cocycles; see Definition 3.2. Using this characterisation, we will prove Theorem C in Subsection 3.4. In Section 4 we prove Theorem D which characterises the set of classes arising as obstructions ω b . In Section 5 we give examples to show that the assumptions of Theorem C are necessary and discuss generalisations. The proof of Proposition 2.3 is postponed to the Appendix in Section 6.

Preliminaries
In this section we recall notation and conventions regarding the (outer) automorphisms in Subsection 2.1. We further recall basic facts on (bounded) cohomology of groups in Subsection 2.2 and quasihomomorphisms by Fujiwara-Kapovich in Subsection 2.3.
2.1. Notation and conventions, Aut and Out Throughout this paper, Roman capitals (A, B) denote groups, lowercase Roman letters (a, b) denote group elements and greek letters (α, β) denote functions. We stick to this notation unless it is mathematical convention to do otherwise. In a group G the identity will be denoted by 1 ∈ G and by 0 ∈ G to stress that G is abelian. The trivial group will also be denoted by "1".
Let N be a group and let Aut(N ) be the group of automorphisms of N . Recall that Inn(N ) denotes the group of inner automorphisms. This is, the subgroup of Aut(N ) whose elements are induced by conjugations of elements in N . There is a map φ : N → Inn(N ) via φ : n → φ n where 3 φ n : g → ngn −1 . Recall that Inn(N ) is a normal subgroup of Aut(N ) and that the quotient Out(N ) = Aut(N )/ Inn(N ) is the group of outer automorphisms of N . It is well-known that there is an exact sequence where Z = Z(N ) denotes the centre of N and all the maps are the obvious ones. We will frequently use the following facts. Let G be a group. Any homomorphism ψ : G → Out(N ) induces an action on Z = Z(N ). This fact is also proved in detail in Subsection 3.1. Moreover, if n 1 , n 2 ∈ N are two elements such that for every g ∈ N , φ n1 (g) = φ n2 (g) then n 1 and n 2 just differ by an element in the centre, i.e. there is z ∈ Z(N ) such that n 1 = zn 2 . This may be seen by the exactness of the above sequence.
Note that δ n restricts to a map C n b (G, V ) → C n+1 b (G, V ) for any n ≥ 0. By abuse of notation we also call this restriction δ n as well.
It is well-known that (C * (G, V ), δ * ) is a cochain complex. The cohomology of G with coefficients in V is the homology of this complex and denoted by H * (G, V ). Similarly (C * b (G, V ), δ * ) is a cochain complex and its homology is the bounded cohomology of G with coefficients in V and denoted by H * b (G, V ). Let W be a normed H-module and let Φ : G → H be a homomorphism. Denote by V the normed abelian group W equipped with G-module structure induced by Φ. We then obtain a map Φ * : Similarly we obtain a map Φ * : . For what follows it will be helpful to work with non-degenerate chains. A map α ∈ C n (G, V ) is called non-degenerate if α(g 1 , . . . , g n ) = 0 whenever g i = 1 for some i = 1, . . . , n. We define | α non-degenerate} and observe that δ * sends non-degenerate maps to non-degenerate maps.
Proposition 2.1. The homology of (N C * (G, V ), δ * ) is H n (G, V ) and the homology of Proof. See Section 6 of [13], where an explicit homotopy between the complexes (N C * (G, V ), δ * ) and (C * (G, V ), δ * ) is constructed. Moreover, one may see that this homotopy preserves bounded maps and hence yields a homotopy between (N C * b (G, V ), δ * ) and (C * b (G, V ), δ * ). Note that the inclusion C * b (G, V ) ֒→ C * (G, V ) commutes with the coboundary operator and hence induces a well defined map c * : , called the comparison map. For a thorough treatment of ordinary and bounded cohomology see [1] and [4] respectively.
If V = Z is equipped with the trivial G-module structure and the standard norm on Z then we see that every class [ω] ∈ ker(c 2 ) may be represented by the coboundary of a map φ : G → Z such that there is a D > 0 such that for all g, h ∈ G |φ(g) + φ(h) − φ(gh)| ≤ D. We call such maps quasimorphisms. Maps ψ : G → Z such that there is a constant D ′ > 0 and a homomorphisms η : G → R such that for all g ∈ G, |φ(g) − η(g)| ≤ D ′ are called trivial quasimorphisms. Many classes of "non-positively curved groups" support non-trivial quasimorphisms. For example acylindrically hyperbolic groups support non-trivial quasimorphisms; see [11]. On the other hand, amenable groups do not support non-trivial quasimorphisms. For a thorough treatment of quasimorphisms see [3].
There are different proposals of how to generalise quasimorphisms φ : G → Z to maps with an arbitrary group as a target φ : G → H. This paper exclusively treats the generalisation of Fujiwara-Kapovich ( [6]). However, we note that there are other generalisations, for example one by Hartnick-Schweitzer ( [10]). The latter are considerably more general than the one we are concerned with; see Subsection 5.2.
The group ∆(σ) < H generated by D(σ) is called the defect group. The map σ : When there is no danger of ambiguity we will write D = D(σ) and ∆ = ∆(σ). This definition is slightly different from the original definition in [6]. Here, the authors required that the set is finite. However, those two definitions may be seen to be equivalent: Proof. We postpone the proof to the Appendix; see Section 6.
We use Definition 2.2 as it is more natural in the context of group extensions. Every set theoretic map σ : G → H with finite image and every homomorphism are quasihomomorphisms for "trivial" reasons. We may also construct different quasihomomorphisms using quasimorphisms φ : G → Z: Let C < H be an infinite cyclic subgroup and let τ : Z → H be a homomorphism s.t. 5 τ (Z) = C. Then it is easy to check that for every quasimorphism φ : Fujiwara-Kapovich showed that if the target H is a torsion-free hyperbolic group then the above mentioned maps are the only possible quasihomomorphisms. To be precise in this case every quasihomomorphism σ : G → H has either finite image, is a homomorphism, or maps to a cyclic subgroup of H; see Theorem 4.1 of [6].
We recall basic properties of quasihomomorphisms. For what follows we use the following convention.
Remark 2.4. If α ∈ Aut(G) and g ∈ G then α g denotes the element α(g) ∈ G. If a ∈ G is an element then a g denotes conjugation by a, i.e. the element aga −1 ∈ G. Sometimes we successively apply automorphisms and conjugations. For example, aα g denotes the element aα(g)a −1 ∈ G.
Proposition 2.5. Let σ : G → H be a quasihomomorphism, let D and ∆ be as above and let H 0 < H be the subgroup of H generated by σ(G). Then ∆ is normal in H 0 . The function φ : G → Aut(∆) defined via φ(g) : a → σ(g) a has finite image and its quotient ψ : Proposition 2.5 may be found in Lemma 2.5 of [6].
so (d, φ) satisfies the identity of the proposition. Rearranging terms we see that Here, for two sets A, B ⊂ H, we write A · B = {a · b ∈ H | a ∈ A, b ∈ B} and A −1 denotes the set of inverses of A. This shows that ∆ is a normal subgroup of H 0 , as D generates ∆, and that φ : G → Aut(∆) has finite image.
Remark 2.6. In light of Proposition 2.5 the extra assumption in Theorem C that the conjugation by the quasihomomorphism induces a finite image in Aut(N ) is natural: Given a short exact sequence 1 → N → E → G → 1 that admits a quasihomomorphic section σ : G → E one may see that 1 → ∆ → E 0 → G → 1 is a short exact sequence where ∆ = ∆(σ) < N and E 0 = σ(G) < E and the map to Aut(∆) has finite image. In fact this assumption is necessary as Example 5.2 shows. 6 Proposition 2.7. Let σ : G → H be a quasihomomorphism. Then the mapσ : G → H defined viaσ is also a quasihomomorphism.
We will use the last proposition to assume that quasihomomorphic sections of extensions satisfy σ(1) = 1.

Extensions and proof of Theorem C
Recall from the introduction that an extension of a group G by a group N is a short exact sequence and that each such extension induces a homomorphism ψ : G → Out(N ). We will recall the construction of such ψ in Subsection 3.1.
In Subsection 3.2 we will define non-abelian cocycles (see Definition 3.2) for group extensions of G by N which induce ψ. Those are certain pairs of functions (e, φ) where e : G × G → N and φ : G → Aut(N ).
We will see that every group extension of G by N inducing ψ gives rise to a non-abelian cocycle (e, φ) in Proposition 3.3. On the other hand every non-abelian cocycle (e, φ) gives rise to an extension 1 → N → E(e, φ) → G → 1; see Proposition 3.4. We will use this correspondence to prove Theorem C in Subsection 3.4. The proof will follow the outline of [1], Chapter VI, 6.

Non-abelian cocyclces
To show Theorem C we will transform the problem of finding all group extensions of G by N which induce ψ to the problem of finding certain pairs (e, φ) called non-abelian cocycles where e : G × G → N and φ : G → Aut(N ) are certain set-theoretic functions.
Then we say that (e, φ) is a non-abelian cocycle with respect to (G, N, ψ).
The idea of studying extensions using these non-abelian cocycles is classical; see Chapter IV, 5.6 of [1]. Here, the author simply calls this a "cocycle condition". In order not to confuse it with the cocycle condition of an ordinary 2-cycle we call it "non-abelian cocycle" with respect to the data for group extensions. Consider Remark 2.4 for the notation of conjugation and action of automorphisms.
Let σ : G → E be the section defined via σ(g) = ν(g)τ (g). Then we see that φ = φ σ . Assume now that the extension is in addition bounded and that φ has finite image. Since the extension is bounded, there is a section τ : G → E which is a quasihomomorphism and such that φ τ : G → Aut(N ) has finite image. By Proposition 2.7 we may assume that τ (1) = 1. We see that we may choose ν : G → N to also have finite image. 8 We claim that the section σ : G → E defined via σ : g → ν(g)τ (g) is a quasihomomorphism. Indeed for any g, h ∈ G we calculate h} which is finite. So all sets on the right hand side are finite and hence σ is a quasihomomorphism. This concludes the proof of Proposition 3.3.
(2) If both φ : G → Aut(N ) and e : G × G → N have finite image then the extension we obtain is bounded (see Definition 1.2).
Proof. Part (1) is classical; see Chapter IV.6 of [1] where such extensions from non-abelian cocycles are implicitly constructed.
For part (2), suppose that both e and φ have finite image then the section σ : G → E(e, φ) is a quasihomomorphism as the defect is just the image of e and, moreover, the map φ σ = φ has finite image. Hence the extension is bounded. This concludes the proof of Proposition 3.4.
For the proof of Theorem C we will need to determine when two non-abelian cocycles correspond up to equivalence to the same group extension. We will need the following statement which is stated, though not proved, at the end of IV.6 in [1].
(1) If (e, φ) is a non-abelian cocycle with respect to (G, N, ψ) then (e ′ , φ) is a non-abelian cocycle with respect to (G, N, ψ) if and only if there is a map c : G × G → Z(N ) = Z satisfying δ 2 c = 0 such that for all g, h ∈ G, e ′ (g, h) = c(g, h) · e(g, h) and for all g ∈ G, c(1, g) = c(g, 1) = 1. Recall that Z(N ) = Z denotes the centre of N .

Proof of Theorem C
We can now prove Theorem C using the correspondence of group extensions with non-abelian cocycles.
Theorem C. Let G and N be groups and suppose that Z = Z(N ), the centre of N , is equipped with a norm · such that (Z, · ) has finite balls. Furthermore, let ψ : G → Out(N ) be a homomorphism with finite image.

There is a class
if and only if E b (G, N, ψ) = ∅ and c 3 (ω b ) = ω is the obstruction of Theorem A. If E b (G, N, ψ) = ∅, then the bijection between the sets H 2 (G, Z) and E(G, N, ψ) described in Theorem A restricts to a bijection between im(c 2 ) ⊂ H 2 (G, Z) and E b (G, N, ψ) ⊂ E(G, N, ψ).
Recall that a normed G-module Z is said to have finite balls if for every K > 0 the set {z ∈ Z | z ≤ K} is finite. We will split the proof into several claims. Claim 3.6 associates to a tuple (G, N, ψ) as in the theorem a function ζ : G × G → N which we then use to define the obstruction class (2). In Claims 3.7 and 3.8 we see that o b is indeed a bounded cocycle and that Z) is independent of the choices made. Finally in Claim 3.9 we see that ω b indeed encodes if (bounded) extensions for the data (G, N, ψ) exist. In Claim 3.10 we construct a bijection Ψ between H 2 (G, Z) (resp. im(c 2 )) and (bounded) extensions.
Let G, N , ψ : G → Out(N ) and Z, · be as in the theorem. Choose a lift φ : G → Aut(N ) of ψ with finite image such that φ(1) = 1.
Claim 3.6. There is a function ζ : G × G → N such that for all g, h ∈ G, n ∈ N , where ζ has finite image in N and for all g ∈ G, ζ(g, 1) = ζ(1, g) = 1.
We now define the obstruction class. Define o b : G × G × G → N so that for all g, h, i ∈ G, and observe that o b necessarily has finite image as both ζ : G × G → N and φ : G → Aut(N ) have finite image. Also, observe that o b (g, h, i) = 1 if one of g, h, i ∈ G is trivial. Proof of Claim 3.7. First we show that o b maps to the centre of N . Observe that for all g, h, i ∈ G and n ∈ N , and hence φ(g) ζ(h, i)ζ(g, hi) and ζ(g, h)ζ(gh, i) induce the same map by conjugation on N and hence just differ by an element of the centre so o b (g, h, i) ∈ Z. Since ζ and φ have finite image, for g, h, i, k ∈ G in two different ways. First observe that Finally, comparing these two terms yields So o b indeed defines a bounded cocycle. This shows Claim 3.7.
is independent of the choices made for ζ and φ. Proof of Claim 3.8. Let φ, φ ′ : G → Aut(N ) be two lifts of ψ as above and choose corresponding functions ζ, ζ ′ : G → N representing the defect of φ and φ ′ as above. There is a finite function ν : G → N with finite image such that φ(g) =ν(g)φ ′ (g) whereν(g) is the element in Inn(N ) ⊂ Aut(N ) corresponding to the conjugation by ν(g). We calculate We see that for every n ∈ N , So ζ(g, h) and ν(g) φ ′ (g) ν(h)ζ ′ (g, h)ν(gh) −1 only differ by an element of the centre. Hence define z(g, h) ∈ Z via ζ(g, h) = z(g, h)ν(g) φ ′ (g) ν(h)ζ ′ (g, h)ν(gh) −1 and note that z : G × G → Z is a function with finite image as all functions involved in its definition have finite image.
It is a calculation to show that o b , the obstruction defined via the choices φ and ζ and o ′ b , the obstruction defined via the choices φ ′ and ζ ′ differ by δ 2 z and hence define the same class in bounded cohomology. This shows Claim 3.8. Z) the obstruction for extensions G by N inducing ψ and denote it by ω b (G, N, ψ) or ω b . We have seen that ω b is a well defined class that depends only on G, N and ψ : G → Out(N ). Next we show that it is an obstruction to (bounded) extensions.  Proof of Claim 3.9. Suppose that c 3 (ω b ) = 0 ∈ H 3 (G, Z). Then there is β ∈ C 2 (G, Z) possibly with unbounded, i.e. infinite image, such that

We call this class [o
for all g, h, i ∈ G. Moreover we may choose β such that for all g ∈ G, β(1, g) = β(g, 1) = 0 by Proposition 2.1 since o b is non-degenerate. Define e : G × G → N via e(g, h) = ζ(g, h)β(g, h) −1 . We will show that (e, φ) is a non-abelian cocycle with respect to (G, N, ψ). Indeed, φ is a lift of ψ which satisfies φ(1) = 1 and for all g ∈ G, e(g, 1) = e(1, g) = 1. Moreover, observe that for all g, h ∈ G and n ∈ N , and hence indeed (e, φ) is a non-abelian cocycle with respect to (G, N, ψ).
By Proposition 3.4, (e, φ) gives rise to an extension of G by N which induces ψ and hence E(G, N, ψ) = ∅.
Analogously, suppose that ω b = 0 in H 3 b (G, Z). Then we may find β ∈ C 3 b (G, Z) satisfying Equation (3), but with bounded i.e. finite image. Hence if we set e(g, h) = ζ(g, h)β(g, h) −1 , we see that e(g, h) has finite image as well, as both ζ and e have. By the above argument (e, φ) is a non-abelian cocycle and, as both e and φ have finite image, (e, φ) gives rise to a bounded extension of (N, G, ψ) by (2) of Proposition 3.4.
On the other hand, suppose that E(G, N, ψ) = ∅. This means that there is some extension 1 → N → E → G → 1 of G by N which induces ψ. By Propositin 3.3, there is a section σ : E → G such that φ σ = φ and then (e σ , φ) is a non-abelian cocycle with respect to (G, N, ψ).
As e σ and ζ have finite image the map β ∈ C 2 (G, Z) defined via e σ (g, h) = ζ(g, h)β(g, h) −1 also has finite image and hence β ∈ C 2 b (G, Z). The above calculations show that o b = δ 2 β and hence ω b = 0 in H 3 b (G, Z). This finishes the proof of Claim 3.9. 13 Now suppose that E b (G, N, ψ) = ∅. then there is an extension 1 → N → E 0 → G → 1 which induces ψ and a section σ 0 : G → E 0 such that φ = φ σ0 and e 0 := e σ0 have finite image and (e 0 , φ) is a non-abelian cocycle with respect to (G, N, ψ).
where α is a non-degenerate representative. Then Ψ is a bijection which restricts to a bijection between im(c 2 ) ⊂ H 2 (G, Z) and E b (G, N, ψ) ⊂ E (G, N, ψ).
Next we show that Ψ is surjective. Let 1 → N → E ′ → G → 1 be any extension of G by N inducing ψ. By Proposition 3.3, there is a section σ ′ : G → E such that φ σ ′ = φ and such that (e ′ , φ) is a non-abelian cocycle with e ′ = e σ ′ . Hence both (e ′ , φ) and (e 0 , φ) are non-abelian cocycles with respect to (G, N, ψ) and by Proposition 3.5 there is a map β ∈ C 2 (G, Z) such that e ′ = β · e 0 and δ 2 β = 0. Then β induces a class and hence Ψ([β]) corresponds to this extension. This shows that Ψ is surjective and hence that Ψ is a bijection. If 1 → N → E ′ → G → 1 is a bounded extension then we may choose a section σ ′ : G → E ′ such that e ′ as above has finite image. Moreover, β as above is bounded as both e ′ and e 0 are. Hence [β] ∈ im(c 2 ) and hence Φ(im(c 2 )) ⊃ E b (G, N, ψ).
This concludes the proof of Theorem C.

The set of obstructions and examples
Theorem C provides a characterisation of non-trivial classes ω b ∈ H 3 b (G, Z), called obstructions. One may wonder which such classes ω b ∈ H 3 b (G, Z) arise in this way. Recall that in the case of general group extensions, every cocycle in H 3 (G, Z) may be realised as such an obstruction: Theorem B. For any G-module Z and any α ∈ H 3 (G, Z) there is a group N with Z = Z(N ) and a homomorphism ψ : G → Out(N ) extensing the G-action on Z such that α = ω(G, N, ψ) in H 3 (G, N, ψ).
For a normed G-module Z with finite balls and a G-action with finite image define the set of , ψ(g) z = g · z, ψ : G → Out(N ) finite}. We refer to the introduction for the definition of F (G, Z) and observe that Theorem D from the introduction may now be restated as follows: 14 Theorem D. Let G be a group and Z be a normed G-module with finite balls and a G-action with finite image. Then . This fully characterises obstructions we obtain in bounded cohomology.
Proof. We have just seen that O b (G, Z) ⊂ F (G, Z), as we may choose ω b in the proof of Theorem C so that it factors through Out(N ) via ψ : G → Out(N ) and Out(N ) is a finite group.
To show F (G, Z) ⊂ O b (G, Z) we need to show that for every finite group M and any class α ∈ H 3 (M, Z) there is a group N and a homomorphism ψ : M → Out(N ) which induces α as a cocycle. We recall a construction from [13]. Working with non-degenerate cocycles (see Subsection 2.2) we may assume that α(1, g, h) = α(g, 1, h) = α(g, h, 1) = 0 for all g, h ∈ G.
Define the abstract symbols g, h for each 1 = g, h ∈ M and set g, 1 = 1, g = 1, 1 = 1 for the abstract symbol 1. Let F be the free group on these symbols and set 1 to be the identity element and set N = Z × F . Define the function φ : M → Aut(N ) so that for g ∈ M the action of φ(g) on F is given by φ(g) h, i = α(g, h, i) g, h gh, i g, hi −1 and so that the action of φ(g) on Z is given by the M -action on Z. A direct calculation yields that for each g ∈ M , the map φ(g) : N → N indeed defines an isomorphism. Here, we need the assumption α(1, g, h) = α(g, 1, h) = α(g, h, 1) = 0. It can be seen that for all n ∈ N and

Examples and Generalisations
We discuss Examples in Subsection 5.1 where we show in particular that the requirements in Definition 1.2 are necessary. Subsection 5.2 discusses possible generalisations of Theorem C.

Examples
The subset E b (G, N, ψ) ⊂ E(G, N, ψ) is generally neither empty nor all of E(G, N, ψ). For any hyperbolic group we have E b (G, N, ψ) = E(G, N, ψ) as the comparison map is surjective ( [14]).
The examples we discuss will use the Heisenberg group H 3 . This group fits into the central extension 1 → Z → H 3 → Z 2 → 1. Elements of the Heisenberg group will be described by [c, z], where c ∈ Z and z ∈ Z 2 . The group multiplication is given by [c 1 , z 1 ] · [c 2 , z 2 ] = [c 1 + c 2 + ω(z 1 , z 2 ), z 1 + z 2 ] where ω(z 1 , z 2 ) = det(z 1 , z 2 ), the determinant of the 2 × 2-matrix (z 1 , z 2  , z), z]. It is well-known that ω generates H 2 (Z 2 , Z) and that ω can not be represented by a bounded cocycle, i.e. the comparison map c 2 : H 2 b (Z 2 , Z) → H 2 (Z 2 , Z) is trivial. Example 5.1. Let G = Z 2 , N = Z and let ψ : G → Out(N ) be the homomorphism with trivial image. The direct product clearly has a quasihomomorphic section that induces a finite map to Aut(N ) and hence E b (G, N, ψ) = ∅. Let Z(N ) = Z be equipped with the standard norm. Note that im(c 2 ) = {0}, for c 2 : H 2 b (Z 2 , Z) → H 2 (Z 2 , Z) the comparison map. By Theorem C, E b (Z 2 , Z, ψ) consists of exactly one element, which is the direct product described above. Note that the Heisenberg is not equivalent to (4). This can be seen as H 3 is not abelian. Hence this extension is not bounded. So in this case Example 5.2. The assumption that the quasihomomorphism σ : G → E has to induce a map φ σ : G → Aut(N ) with finite image may seem artificial, as the induced homomorphism ψ : G → Out(N ) already has finite image. However it is necessary as the following example shows.
Consider Pick an isomorphism φ : Z 2 → Inn(H 3 ). We may construct the extension where H 3 ⋊ φ Z 2 denotes the semi-direct product. and observe that the action of Z 2 on the centre of H 3 is trivial as the automorphisms are all inner.
Suppose that σ : G → H is a quasihomomorphism in the sense of Definition 2.2. We start by noting the following easy property. Claim 6.1. Let σ : G → H be a quasihomomorphism with defect group ∆ and let A ⊂ ∆ be a finite subset of ∆. Then the set { σ(g) A | g ∈ G} is also a finite subset of ∆.
Proof. By Proposition 2.5, the set of automorphisms {a → σ(g) a | g ∈ G} ⊂ Aut(∆) is finite. Hence we see that the set { σ(g) A | g ∈ G} is the image of a finite set of ∆ under finitely many automorphisms of ∆ and hence a finite subset of ∆.
Using the claim again we see that is finite and hence that D(σ) = {σ(h) −1 σ(g) −1 σ(gh) | g, h ∈ G} ⊂ D 2 σ(h −1 )σ(g −1 )σ((gh) −1 ) ⊂ D 2 D soD(σ) is indeed a finite set. This shows that any quasihomomorphism in the sense of Definition 2.2 is a quasihomomorphism in the sense of [6]. Now assume that σ : G → H is a map such that the setD =D(σ) is finite and let∆ be the group generated byD.
Just as before we have the following claim: Claim 6.2. Let f : G → H be a map such thatD =D(f ) is finite and let∆ be the group generated byD. If A ⊂∆ if a finite subset of∆ then the set is also a finite subset of∆.
Proof. This follows from the same argument as for Claim 6.1 using Lemma 2.5 of [6] instead of Proposition 2.5.