The spectrum of simplicial volume

New constructions in group homology allow us to manufacture high-dimensional manifolds with controlled simplicial volume. We prove that for every dimension bigger than 3 the set of simplicial volumes of orientable closed connected manifolds is dense in $\mathbb{R}_{\geq 0}$. In dimension 4 we prove that every non-negative rational number is the simplicial volume of some orientable closed connected 4-manifold. Our group theoretic results relate stable commutator length to the $l^1$-semi-norm of certain singular homology classes in degree 2. The output of these results is translated into manifold constructions using cross-products and Thom realisation.


Introduction
The simplicial volume M of an orientable closed connected (occ) manifold M is a homotopy invariant that captures the complexity of representing fundamental classes by singular cycles with real coefficients (see Section 2 for a precise definition and basic terminology). Simplicial volume is known to be positive in the presence of enough negative curvature [Gro82,Thu97,IY82,LS06] and known to vanish in the presence of enough amenability [Gro82,Iva85,Yan82,BCL18]. Moreover, it provides a topological lower bound for the minimal Riemannian volume (suitably normalised) in the case of smooth manifolds [Gro82].
Until now, for large dimensions d, very little was known about the precise structure of the set SV(d) ⊂ R ≥0 of simplicial volumes of occ d-manifolds. The set SV(d) is countable and closed under addition (Remark 2.3). However, the set of simplicial volumes is fully understood only in dimensions 2 and 3 with SV(2) = N [4]  This reveals that there is a gap of simplicial volume in dimensions 2 and 3: For d ∈ {2, 3} there is a constant C d > 0 such that the simplicial volume of an occ d-manifold either vanishes or is at least C d . It was an open question [Sam99,p. 550] whether such a gap exists in higher dimensions. For example, until now the lowest known simplicial volume of an occ 4-manifold has been 24 [BK08] (Example 2.6).
In the present paper, we show that dimensions 2 and 3 are the only dimensions with such a gap.

Introduction
Theorem A (no-gap; Section 8.2). Let d ≥ 4 be an integer. For every > 0 there is an orientable closed connected d-manifold M such that 0 < M ≤ . Hence, the set of simplicial volumes of orientable closed connected d-manifolds is dense in R ≥0 .
In dimension 4, we get the following refinement of Theorem A.
Theorem B (rational realisation; Section 8. 3). For every q ∈ Q ≥0 there is an orientable closed connected 4-manifold M q with M q = q.

Method
We first compute the l 1 -semi-norm of certain integral 2-classes in finitely presented groups by relating these semi-norms to stable commutator length.
To formulate this connection, we recall some definitions. For a group G and a class α ∈ H d (G; R), the l 1 -semi-norm α 1 of α is the semi-norm induced by the l 1 -norm of chains in the singular chain complex of any model of BG (Section 2.1). The class α is integral if it lies in the image under the change of coefficient map map induced by Z → R.
For an element g ∈ [G, G] in the commutator subgroup of G, the commutator length cl G (g) of g is the minimal number of commutators in G needed to express g as their product. The stable commutator length (scl) of g is the limit scl G (g) := lim n→∞ cl G (g n )/n. Stable commutator length is now wellunderstood for many classes of groups thanks largely to Calegari and his coauthors [Cal09a].
Theorem C (Corollary 6.16). Let G be a finitely presented group with H 2 (G; R) ∼ = 0 and let g ∈ [G, G] be an element of infinite order. Then there is a finitely presented group D(G, g) and an integral class α g ∈ H 2 (D(G, g); R) such that α g 1 = 8 · scl G g.
We apply Theorem C to the universal central extension E of Thompson's group T . Recall that T is the group of piecewise linear homeomorphisms of the circle with dyadic breakpoints and whose slopes are an integer power of 2. In Propostion 5.1, we show that the universal central extension E of T is a finitely presented group with H 2 (E; R) ∼ = 0 and that every non-negative rational number may be realised by the stable commutator length of some element in E. Using Theorem C this shows: Theorem D (Corollary 6.17). For every q ∈ Q ≥0 there is a finitely presented group G q and an integral class α q ∈ H 2 (G q ; R) such that α q 1 = q. In particular, for every > 0 there is a finitely presented group G and an integral class α ∈ H 2 (G ; R) such that 0 < α ≤ .
We can now take cross-products in homology to obtain integral classes in degree greater than 3 with crude norm control. Applying a normed version of Thom realisation (Theorem 8.1) then proves Theorem A.
In dimension 4, we refine this construction by taking products with surfaces and using exact computation of the product norm. This generalises a result of Bucher [BK08]. Theorem B will follow from these computations.
In particular, we establish the following connection between stable commutator length and simplicial volume in dimension 4: Theorem F (Corollary 8.3). Let G be a finitely presented group that satisfies H 2 (G; R) ∼ = 0 and let g ∈ [G, G] be an element in the commutator subgroup. Then there is an orientable closed connected 4-manifold M g with M g = 48 · scl G g.

Organisation of this article
Sections 2, 3, and 4 recall basic properties and known results on simplicial volume, bounded cohomology and stable commutator length, respectively.
In Section 5 we compute scl on the universal central extension E of Thompson's group T (Proposition 5.1). This will be used in Section 6 to construct integral 2-classes with controlled l 1 -semi-norms. There we also show Theorems C and D.
In Section 7 we get the refinement for dimension 4 in group homology: We compute the l 1 -semi-norm of cross-products of general 2-classes with certain Euler-extremal 2-classes (Theorem 7.1). As a corollary we obtain Theorem E.
All manifolds constructed in this article will arise via a suitable version of Thom's realisation theorem in Section 8. This allows us to manufacture manifolds with controlled simplicial volume and to prove Theorems A, B, and F.
A discussion of related problems may be found in Section 8. 4. In the Appendix A, we discuss a construction for possible transcendental values of simplicial volume in occ 4-manifolds.
On the one hand, simplicial volume clearly is a topological invariant of (orientable) compact manifolds that is compatible with mapping degrees. On the other hand, simplicial volume is related in a non-trivial way to Riemannian volume, e.g., in the case of hyperbolic manifolds [Gro82,Thu97]. Therefore, simplicial volume is a useful invariant in the study of rigidity properties of manifolds.
Basic examples of simplicial volumes are listed in Examples 2.4, 2.5, and 2.6. In addition to geometric arguments, a key tool for working with simplicial volume is bounded cohomology (see Proposition 3.4 below).

Simplicial volume in low dimensions and gaps
We collect the low-dimensional examples of simplicial volume as stated in the introduction. Recall that for an integer d we define SV(d) ⊂ R ≥0 via SV(d) := M M is an orientable closed connected d-manifold . Remark 2.3. As there are only countably many homotopy types of orientable closed connected (occ) manifolds [Mat65], the set SV(d) is countable for every d ∈ N.
The set SV(d) is also closed under addition. For d ≥ 3, this follows from the additivity of simplicial volume under connected sums [Gro82][Fri17, Corollary 7.7] and for d = 2 this follows from the explicit computation of SV(2) as seen in Example 2.4.

2.3
The l 1 -semi-norm in degree 2 As classes in degree 2 will play an important role in our constructions, we collect some basic properties concerning the l 1 -semi-norm in degree 2.

Simplicial volume of products
Remark 2.8. Let X be a path-connected topological space, let α ∈ H 2 (X; Z), and let α R ∈ H 2 (X; R) be the image of α under the change of coefficients map Z → R. Then the description of α R 1 from Proposition 2.7 simplifies as follows: We have where Σ(α) is the class of all pairs (f, Σ) consisting of an orientable closed connected surface Σ of genus g(Σ) ≥ 1 and a continuous map f : In Section 6, we will relate l 1 -semi-norms of relative classes in degree 2 to filling invariants and stable commutator length.

Simplicial volume of products
We recall basic results on l 1 -semi-norms of homological cross-products.
Proposition 2.9. Let X, Y be topological spaces, let m, n ∈ N, and let α ∈ H m (X; R), β ∈ H n (Y ; R). Then the cross-product α × β ∈ H m+n (X × Y ; R) satisfies Proof. The lower estimate follows from the duality principle (Proposition 3.4) and an explicit description of the cohomological cross-product (in bounded cohomology), the upper estimate follows from an explicit description of the homological cross-product [Gro82][BP92, Theorem F.2.5] (this classical argument works also for general homology classes, not only for fundamental classes of manifolds).
However, in general, it seems to be a hard problem to compute the exact values of l 1 -semi-norms of products. One of the few known cases are products of two orientable closed connected surfaces, whose simplicial volumes have been computed by Bucher: Theorem 2.10 ([BK08, Corollary 3]). Let Σ g , Σ h be orientable closed connected surfaces of genus g, h ∈ N ≥1 . Then We will generalise this theorem in Section 7. For now, let us note that in combination with the description of the l 1 -semi-norm in degree 2 in terms of surfaces, we obtain the following general, improved, upper bound: Corollary 2.11. Let X and Y be path-connected topological spaces and let α ∈ H 2 (X; R), β ∈ H 2 (Y ; R). Then Proof. We use the description of α 1 and β 1 from Proposition 2.7. Let be surface presentations of α and β as in Proposition 2.7. Then Therefore, applying the triangle inequality, functoriality (Remark 2.1), and Theorem 2.10, shows that By Proposition 2.7, taking the infimum over all such surface representations of α and β, we obtain α × β 1 ≤ 3/2 · α 1 · β 1 .

Bounded cohomology
Bounded cohomology of discrete groups and topological spaces was first systematically studied by Gromov [Gro82]. Gromov established the fundamental properties of bounded cohomology using so-called multicomplexes. Later, Ivanov developed a more algebraic framework via resolutions [Iva85,Iva17]. The reference to this introduction is the recent book by Frigerio [Fri17]. Having applications to stable commutator length and the l 1 -semi-norm in mind we will only define bounded cohomology for trivial real and integer coefficients. Sections 3.1, 3.2 and 3.3 discuss the (relationships between) bounded cohomology of groups and topological spaces. In Section 3.4 we state the duality principle, which allows us to compute the l 1 semi-norm. In Section 3.6 we define the Euler class.

Bounded cohomology of groups
Let V be R or Z and let G be a group. We will define the bounded cohomology H n b (G; V ) of G using the homogeneous resolution. There is also an inhomogeneous resolution, which is useful in low dimensions. We use this resolution only in Section 5 for central extensions and refer to the literature [Fri17, Chapter 1.7] for the definition.
Let C n (G; V ) := map(G n+1 , V ) be the set of set-theoretic maps from G n+1 to V . The group G acts on C n (G; V ) via g · φ(g 0 , . . . , g n ) = φ(g −1 · g 0 , . . . , g −1 · g n ). We denote by C n (G; V ) G the subset of elements in C n (G; V ) that are

Bounded cohomology of spaces
invariant under this action. Let · ∞ be the l ∞ -norm on C n (G; V ) and let C n b (G; V ) be the corresponding subspaces of bounded functions. Define the well-known coboundary maps δ n : where α(g 0 , . . . , g i , . . . , g n+1 ) means that the i-th coordinate is omitted. Then δ n restricts to a map Bounded cohomology is functorial in both the group and the coefficients.

Bounded cohomology of spaces
Let X be a topological space and let S n (X) be the set of singular n-simplices in X. Moreover, let C n (X; V ) be the set of maps from S n (X) to V . For an element α ∈ C n (X; V ) we set and let C n b (X; V ) ⊂ C n (X; V ) be the subset of elements that are bounded with respect to this norm. Let δ n : C n b (X; V ) → C n+1 b (X; V ) be the restriction of the singular coboundary map to bounded cochains. Then the bounded cohomology H n b (X; V ) of X with coefficients in V is the cohomology of the complex (C • b (X; V ), δ • ) and denoted by H n b (X; V ). For α ∈ H n b (X; V ) we define and observe that · is a semi-norm on H n b (X; V ). The bounded cohomology of spaces is also functorial in both spaces and coefficients.

Relationship between bounded cohomology of groups and spaces
Analogously to ordinary group cohomology, also bounded cohomology of groups may be computed using classifying spaces (and thus, we will freely switch between these descriptions). Remarkably, this statement holds true much more generally: every topological space with the correct fundamental group can be used to compute bounded cohomology of groups; moreover, bounded cohomology ignores amenable kernels.

Duality
Bounded cohomology of groups and spaces may be used to compute the l 1 -seminorm of homology classes. For what follows, let ·, · : H n b (X; V ) × H n (X; V ) → V be the map given by evaluation of cochains on chains. . Let X be a topological space and let α ∈ H n (X; R). Then Moreover, the supremum is achieved.
Observe that alt n (α) ∞ ≤ α ∞ . The subcomplex of alternating cochains is denoted by C n b,alt (G; V ). It is well-known that one can compute real bounded cohomology using alternating cochains:

Euler class and rotation number
Proposition 3.6 ([Fri17, Proposition 4.26]). Let G be a group. The complex C • b,alt (G, R) isometrically computes the bounded cohomology with real coefficients. Moreover, for every α ∈ C n b (G, R) the cocycle alt n b (α) represents the same class as α in H n b (G; R).
All classes that we define via Eu will be independent of ξ.
Remark 3.7. For Euler classes (and the orientation cocycle) we will use the following notation: Capital letters (Eu) denote cocycles and lower case letters (eu) denote classes. The classes eu are the ones represented by Eu in the corresponding cohomology groups. If a group G acts on the circle by ρ : G → Homeo + (S 1 ), and α is a class or a cocycle defined on Homeo + (S 1 ) then ρ * α will be the pullback of α via ρ. If Γ < Homeo + (S 1 ) is a subgroup of Homeo + (S 1 ), then we will denote the restriction of a class or a cocycle α to Γ by Γ α. Hence, for example, Γ eu R b ∈ H 2 b (Γ; R) denotes the restriction of the real bounded Euler class to Γ. Let G be a group with a circle action ρ : G → Homeo + (S 1 ). Then ρ * eu Z ∈ H 2 (G; Z) is called the Euler class associated to the action ρ. The Euler class induces a central extension of G by Z, the associated Euler extension G. This group has the following explicit description. It is the group defined on the set Z × G with multiplication (z, g) · (z , g ) = (z + z + ρ * Eu(1, g, g · g ), g · g ). Euler extensions are useful for constructing groups with controlled stable commutator length; see Section 5. We note that Example 3.8. Let g ∈ N ≥2 and let Σ g be an oriented closed connected surface denotes the fundamental class and Γ g = π 1 (Σ g ). Then Γ g induces a natural action on its boundary. By identifying ∂Γ g ∼ = S 1 , we obtain a circle action ρ : Γ g → Homeo + (S 1 ) and is an extremal cocycle for the fundamental class [Σ g ] R . Indeed, it is the renormalised volume cocycle of ideal simplices in H 2 ; see [BK08].

Stable commutator length
In recent years the topic of stable commutator length (scl) has seen a vast developemet thanks largely to Calegari and his coauthors [Cal09a]. In this section, we will only give a brief overview of scl. The definition and basic properties will be given in Section 4.1. A useful tool to compute scl is Bavard's duality theorem, described in Section 4.2. We discuss examples and general properties of scl in Section 4.3.

Definition and basic properties
For a group G let G be its commutator subgroup. The commutator length cl G (g) of an element g ∈ G is defined as n .
If g 1 , . . . , g m ∈ G are such that g 1 · · · g m ∈ G , we will call g 1 + · · · + g m a chain and define the corresponding (stable) commutator length on chains by If ϕ : G → H is a group homomorphism, then scl G (g) ≥ scl H (ϕ(g)) for all g ∈ G ; the analogous result holds for chains. In particular, scl is invariant under automorphisms, whence under conjugation. Thus, scl on single chains agrees with the usual definition of stable commutator length.
Stable commutator length has the following geometric interpretation: If X is a connected topological space and γ : S 1 → X is a loop, then the stable commutator length of the associated element [γ] ∈ π 1 (X) measures the least complexity of the surface needed to bound γ (we will not use this interpretation in this paper). In Section 6.2, we will describe yet another interpretation of scl, namely as a topological stable-filling invariant.

Bavard's duality theorem and bounded cohomology
The smallest such C is called the defect of φ and is denoted by D(φ). A quasimorphism φ is homogeneous if in addition we have that φ(g n ) = n · φ(g) for all g ∈ G, n ∈ Z. Every quasimorphism φ : G → R is in bounded distance to a unique homogeneous quasimorphismφ : G → R, defined by settinḡ Analogously to the duality principle (Proposition 3.4) we may compute scl using homogeneous quasimorphisms: Theorem 4.1 (Bavard's duality theorem; [Bav91]). Let G be a group and let g ∈ G . Then where the supremum is taken over all homogeneous quasimorphisms φ : G → R. Moreover, this supremum is achieved by an extremal quasimorphism.
Remark 4.2. (Homogeneous) quasimorphisms are intimately related to second bounded real cohomology. Using the inhomogeneous resolution, it can be seen that the kernel of c 2 corresponds to the space of homogeneous quasimorphisms modulo Hom(G, R). It follows then from Bavard's dualtiy theorem that the comparison map c 2

Examples
We collect some known results for stable commutator length.
In Sections 6 and 8 we will promote scl in a finitely presented group G to the simplicial volume of manifolds in higher dimension. For this we need to assert that H 2 (G; R) vanishes. Thus, we will have a particular emphasis on this condition in the examples.

Vanishing
An element g ∈ G may satisfy that scl G (g) = 0 for "trivial" reasons, such as if g is torsion or if g is conjugate to its inverse. There are many classes of groups where -besides these trivial reasons -stable commutator length vanishes on the whole group. Recall that this is equivalent to the injectivity of the comparison map c 2 G : H 2 b (G; R) → H 2 (G; R). Examples include: • amenable groups: This follows from the vanishing of H 2 b (G; R) for every amenable group G by a result of Trauber [Gro82], • irreducible lattices in semisimple Lie groups of rank at least 2 [BM02], and

Non-abelian free groups
In contrast, Duncan and Howie [DH91] showed that every element g ∈ F \{e} in the commutator subgroup of a non-abelian free group F satisfies scl F (g) ≥ 1/2. In a sequence of papers [Cal09b,Cal11] Calegari showed that stable commutator length is rational in free groups and that every rational number mod 1 is realised as the stable commutator length of some element in the free group. Moreover, he gave an explicit, polynomial time algorithm to compute stable commutator length in free groups. This revealed a surprising distribution of those values. We note that these results generalise to free products of cyclic groups and that all these groups G satisfy H 2 (G; R) ∼ = 0.

Gaps and groups of non-positive curvature
A group G has a gap in scl if there is a constant C > 0 such that for every group element g, we have scl G (g) ≥ C unless scl G (g) = 0 for "trivial" reasons such as torsion or if g is conjugate to its inverse.
In the previous example, we already have seen that non-abelian free groups have a gap in stable commutator length of 1/2. This result has recently been generalised to right-angled Artin groups [Heu19]. Many classes of non-positively curved groups have a gap in scl, though this gap may not be uniform in the whole class of groups. Prominent examples include hyperbolic groups [CF10], mapping class groups [BBF16], free products of torsion free groups [Che18] and amalgamated free products [CFL16, Heu19, CH].

Hyperelliptic mapping class groups
Let g ∈ N, let ι ∈ M g be the mapping class of a hyperelliptic involution of the orientable closed connected surface Σ g of genus g, and let . We now let g ≥ 2. Let t ∈ H g be a Dehn twist about a ι-invariant non-separating curve on Σ g . Then we have

The universal central extension of Thompson's group T
Thompson's group T was introduced in 1965 by Richard Thompson as the first example of an infinite but finitely presented simple group. It is the subgroup of PL + (S 1 ) with dyadic breakpoints and where each derivative -if defined -is an integer power of 2 (here, we identify R/Z ∼ = S 1 ) [CFP96]. Stable commutator length on Thompsons's Group T vanishes [Cal09a, Chapter 5], but interesting values for stable commutator length arise on the central extensions of T and its generalisations associated to the Euler class [Zhu08] (Section 3.6).
In this section, we extend these results about stable commutator length on these extensions to the universal central extension E of T .
Proposition 5.1. The universal central extension E of Thompson's group T is finitely presented and satisfies that H 1 (E; Z) ∼ = 0 ∼ = H 2 (E; Z). For every nonnegative rational number q ∈ Q ≥0 , there is an element e q ∈ E with scl E (e q ) = q.

The Euler central extension of Thompson's group T
In Section 5.1, we recall results of Zhuang [Zhu08], which describe stable commutator length on the central extension T of T associated to the Euler class. Using information on the (bounded) 2-cohomology of Thompson's group T (Section 5.2), we reduce stable commutator length on E to stable commutator length on T and show Proposition 5.1 (Section 5.3).

The Euler central extension of Thompson's group T
We recall the connection between stable commutator length and rotation number. This connection has been established by Barge and Ghys [BG88] and has been used by Zhuang [Zhu08] to construct finitely presented groups with transcendental stable commutator length (see Appendix A). Zhu08]). Let T be the central extension of Thompson's Group T associated to the Euler class T eu Z ∈ H 2 (T ; Z). Then there is a homogeneous quasimorphism rot : T → R of defect 1, called rotation number, that generates the space of homogeneous quasimorphisms. Hence, for all t ∈ T , The rotation number is well studied and has a geometric meaning [BFH16]. Hence, one obtains the full spectrum of stable commutator length for T . Proof. Ghys and Sergiescu [GS87] showed that the rotation number on T is rational. Moreover, it is well known that every rational number is realised as such a rotation number.
However, Ghys and Sergiescu [GS87] showed that H 2 ( T ; Z) ∼ = Z. Thus, we cannot apply Theorem C to the group T .

(Bounded) 2-cohomology of Thompson's group T
The cohomology of Thompson's group T was computed by Ghys and Sergiescu [GS87]. Ghys and Sergiescu showed that the 2-cohomology H 2 (T ; Z) is generated by the Euler cocycle T eu Z (see Section 3.6) and another class α, which has the following combinatorial description.
For a function φ : S 1 → R that admits limits on both sides, let φ(x + ) be the right and let φ(x − ) be the left limit at a point x ∈ S 1 . In this case, set Definition 5.4 (discrete Godbillon-Vey cocycle; [GS87, Theorem E]). The discrete Godbillon-Vey cocycle gv : where the (finite) sum runs over all x ∈ S 1 that are breakpoints of v, u or u • v. The map gv is an inhomogeneous cocycle. In this section only we will use inhomogeneous cocycles as they are better to work with in the context of central extensions; the precise definition can for instance be found in Frigerio's book [Fri17, Chapter 1.7]. For what follows we will also need to compute the bounded cohomology of T in degree 2.
Proposition 5.6. The class α ∈ H 2 (T ; Z) from Theorem 5.5 cannot be represented by a bounded cocycle, i.e., α is not in the image of the comparison map H 2 b (T ; Z) → H 2 (T ; Z). In particular, we have that generated by the Euler class.
Proof. Note that it is enough to show the unboundedness statement for [gv] as 2 · α = [gv] (Theorem 5.5). We will show the proposition by evaluating gv on the subgroup Z 2 ∼ = a, b T ⊂ T , where a and b are the elements depicted in Figure 1.
Claim 5.7. The cocycle gv restricts on a, b T to a cocyle representing a generator of H 2 (Z 2 ; Z).
Proof of Claim 5.7. This claim is implicitly stated in the work of Ghys and Sergiescu [GS87, proof of Lemma 4.6]. For the convenience of the reader we provide an explicit proof here.
for any x ∈ [0, 1/2) and that This way we see that Similarly, we see that We moreover calculate Putting the above calculations together, we can now compute the restriction of gv to a, b T . For all i, j, i , j ∈ Z we see that where f 0 (i, j) := f (i) and g 0 (i, j) := g(j). Hence, gv restricted to a, b T represents a generator of H 2 (Z 2 ; Z). This proves Claim 5.7.
It is well-known that non-trivial elements of H 2 (Z 2 ; Z) cannot be represented by a bounded cocycle (Z 2 is amenable). Hence, also [gv] cannot be represented by a bounded cocycle, which proves the first part of Proposition 5.6.
Stable commutator length vanishes on T (Example 4.3.1) and so the comparison map c 2 T : H 2 b (T ; R) → H 2 (T ; R) is injective (Remark 4.2). We now assume for a contradiction that λ · T eu R + µ · α ∈ H 2 (T ; R) lies in the image of the comparison map c 2 T : H 2 b (T ; R) → H 2 (T ; R) and µ = 0. Then λ· T eu R +µ·α restricts to a non-trival class on a, b T and generates H 2 (Z 2 ; R). This is a contradiction as these classes are not bounded. Hence, the only classes in the image of c 2 T are multiples of T eu R . We conclude that generated by the Euler class. This completes the proof of Proposition 5.6.

Proof of Proposition 5.1
We will now prove Proposition 5.1 by explicitly describing the quasimorphisms on E and invoking Bavard duality (Theorem 4.1). For a more homological argument -and a possible generalisation -we refer to Appendix A.
Proof of Proposition 5.1. As E is an extension of a finitely presented group by a finitely presented group it is finitely presented itself. As T is a simple group [CFP96], it is in particular perfect. Recall that the universal central extension E of a perfect group always satisfies that H 1 (E; Z) ∼ = 0 and H 2 (E; Z) ∼ = 0 [Wei94, Chapter 6.9].
We will now show that scl E and scl T have the same image in R: In view of Theorem 5.5, the universal central extension group of T can be explicitly described as the group on the set Z 2 × T with group multiplication where [ T Eu] = T eu Z ∈ H 2 (T ; Z) is the inhomogeneous Euler cocycle and A is an inhomogeneous cocycle representing α ∈ H 2 (T ; Z). Let φ : E → R be a homogeneous quasimorphism on E. Then φ restricts to a homomorphism on the central subgroup Z 2 . Let λ 1 , λ 2 ∈ R be such that We define ψ : T → R via t → φ(0, 0, t). As φ is a quasimorphism the term is uniformly bounded in t, t ∈ T . Using that T Eu is bounded, we see that is uniformly bounded as well. By Proposition 5.6 this implies that λ 2 = 0. Hence, for every i, j ∈ Z, t ∈ T we have that where we used that (Z, 0, 0) ⊂ E is central and φ is homogeneous. Applying Bavard's duality theorem, we conclude that scl E (i, j, t) = scl E (0, j, t).
On the other hand, let φ : E → R be an extremal homogeneous quasimorphism to (0, j, t). Then the above calculations show that φ induces a quasimorphism on T with the same defect and hence by using again Bavard duality. We conclude that for every i, j ∈ Z and t ∈ T . Corollary 5.3 asserts that every non-negative rational number gets realised as the stable commutator length of some element in T . This finishes the proof of Proposition 5.1.

Fillings
Stable commutator length can be interpreted as a homological filling norm (Section 6.2). After recalling the basic notions and properties, we will use this interpretation to compute the l 1 -semi-norm of classes related to decomposable relators and thus prove Theorem C (Section 6.3). This will allow us to establish the group-theoretic version of the no-gap theorem (Theorem D) . Moreover, we will explain how the simplicial volume of manifolds can also be viewed as a filing norm (Section 6.4).

Stable filling norms
We first recall the stable filling norm for the bar complex. We will then extend this notion to topological spaces and higher degrees. For a group G, the bar complex C • (G; R) (computing H • (G; R)) has the following form in low degrees: We have C 1 (G; R) = R[G] and ∂ 1 := 0 as well as C 2 (G; R) = R[G] 2 and Moreover, the chain modules of C • (G; R) are endowed with the l 1 -norm corresponding to the bar bases.
Definition 6.1 ((stable) filling norm). Let G be a group.
Notice that the limit in the definition of the stable filling norm indeed exists [Cal09a,p. 34].
For the generalisation to topological spaces, we replace group elements by loops (or maps from simplicial spheres) and we replace taking powers of group elements by composition with self-maps of spheres of the corresponding degree.
Definition 6.2 (topological (stable) filling norms). Let d ∈ N, let X be a topological space, and let σ : ∂∆ d → X be continuous.
• If c ∈ ∂(C d (X; R)), the filling norm of c is defined as • The filling norm of σ is then defined as is the canonical singular cycle associated with σ.
• The stable filling norm of σ is defined as where for n ∈ N, we write w n : ∂∆ d → ∂∆ d for "the" standard self-map of ∂∆ d ∼ = S d−1 of degree n and σ[n] := σ • w n .
• If m ∈ N and σ 1 , . . . , σ m : ∂∆ d → X are continuous maps, then we define Remark 6.3 (existence of the stabilisation limit). The limits in the situation of the definition above indeed exist: For notational convenience, we only prove the existence in the case of sfill X σ; the general case can be proved in the same way (with additional indices). The argument is similar to the one for the stable filling norm in the bar complex. The only complication is that, in order to compare different "powers", we will need to use the uniform boundary condition for C • (∂∆ d ; R). Because π 1 (∂∆ d ) is amenable, there exists a constant K ∈ R >0 with the following property [MM85,FL19]: For every z ∈ ∂(C d (∂∆ d ; R)) there is a b ∈ C d (∂∆ d ; R) with z = ∂b and |b| 1 ≤ K · |z| 1 .
If n, m ∈ N, then the chains c wn + c wm and c wn+m are homologous in the complex C • (∂∆ d ; R) (because deg w n + deg w m = n + m = deg w n+m ). Thus, there exists a chain b n,m ∈ C d (∂∆ d ; R) such that c wn+m − c wn − c wm = ∂b n,m and |b n,m | 1 ≤ K · 3 · (d + 1).
Hence, for every continuous map σ : ∂∆ d → X we obtain Now elementary analysis shows that the limit lim n→∞ 1/n · fill X σ[n] does exist.
Remark 6.4 (change of the self-maps). The map w n is only unique up to homotopy, but homotopic choices for w n lead to the same stable filling norm; this can be seen using the uniform boundary condition as in the proof of the existence of the stable filling limits (Remark 6.3). Therefore, this ambiguity will be of no consequence for us.
Remark 6.5 (change of the singular models). In the situation of Definition 6.2, we could choose other singular cycle models of σ than c σ : For the second equality, we use that |C d (σ[n]; R)(b )| 1 ≤ |b | 1 holds for all n ∈ N (so that the difference in norm is negligible when taking n → ∞).
This can be seen as follows: The standard constructions produce chain maps ϕ : C • (G; R) → C • (X; R) (choosing paths in X for each group element and inductively filling the simplices) and ψ : C • (X; R) → C • (G; R) (choosing a set-theoretic fundamental domain D for the deck transformation action on X and looking at the translates of D that contain the vertices of the lifted simplices) with the following properties: • ϕ • ψ id C•(X;R) through a chain homotopy h that is bounded in every degree, • ϕ ≤ 1 and ψ ≤ 1.

Stable commutator length as filling invariant
The fact that every commutator consists of four pieces has the following generalisation in terms of filling norms: Lemma 6.7 (scl as filling invariant). Let G be a group, let m ∈ N, and let r 1 , . . . , r m ∈ G . Then scl G (r 1 + · · · + r m ) = 1 4 · sfill G (r 1 + · · · + r m ). Furthermore, as Calegari [Cal08] puts it: "One can interpret stable commutator length as the infimum of the L 1 norm (suitably normalized) on chains representing a certain (relative) class in group homology." We will prove this statement in Corollary 6.9 as a special case of the following generalisation: Proposition 6.8 (relative l 1 -semi-norm as filling invariant). Let Z be a CWcomplex, let m ∈ N >0 , d ∈ N ≥2 , let ∂Z ⊂ Z be a subspace that is homeomorphic to m ∂∆ d and such that the inclusions σ 1 , . . . , σ m : ∂∆ d → Z of the m components of ∂Z into Z are π 1 -injective (this is automatic if d ≥ 3).
Corollary 6.9 (scl as relative l 1 -semi-norm). Let G be a group with H 2 (G; R) ∼ = 0, let m ∈ N, let r 1 , . . . , r m ∈ G be elements of infinite order, and let X be a model of BG. Let be the mapping cylinder associated with (loops γ 1 , . . . , γ r in X representing) the elements r 1 , . . . , r m , and let ∂Z := m S 1 × {1} ⊂ Z. Then there exists a unique relative homology class β ∈ H 2 (Z, ∂Z; R) whose boundary class ∂β is the fundamental class of m S 1 ; the class β satisfies Proof. The long exact homology sequence of the pair (Z, ∂Z) shows that the connecting homomorphism ∂ : H 2 (Z, ∂Z; R) → H 1 (∂Z; R) is an isomorphism (by hypothesis, H 2 (Z; R) ∼ = H 2 (X; R) ∼ = H 2 (G; R) ∼ = 0, and the inclusion ∂Z → Z induces the trivial homomorphism on H 1 ( · ; R) because r 1 , . . . , r m are in the commutator subgroup of G). This shows the existence of β.
Proof. Clearly, S S 1 is a model of BF (S) and H 2 (F (S); R) ∼ = 0. Therefore, we can apply Corollary 6.9.

Decomposable relators
The filling view allows us to compute the l 1 -semi-norm for certain classes in degree 2 associated to "decomposable relators" in terms of stable commutator length. Let us first describe these homology classes: Setup 6.11 (decomposable relators I). Let G 1 and G 2 be groups that satisfy H 2 (G 1 ; R) ∼ = 0 and H 2 (G 2 ; R) ∼ = 0 and let r 1 ∈ G 1 , r 2 ∈ G 2 . We then consider the glued group where the amalgamation homomorphisms Z → G 1 and Z → G 2 are given by r 1 and r −1 2 , respectively.

Decomposable relators
Associated with this situation, there is a canonical homology class α ∈ H 2 (D(G 1 , G 2 , r 1 , r 2 ); R): Let X 1 and X 2 be classifying spaces for G 1 and G 2 , respectively. We consider the cylinder spaces for the relators r 1 and r 2 , respectively. Then is a CW-complex such that the canonical maps Z 1 → P and Z 2 → P induce an isomorphism π 1 (P ) ∼ = D(G 1 , G 2 , r 1 , r 2 ) =: G.
Remark 6.12 (integrality of the canonical class). In the situation of Setup 6.11, the canonical homology class α is integral: It suffices to show that α ∈ H 2 (P ; R) is integral. Comparing the long exact sequences of (Z 1 , ∂Z 1 ) with Z-and R-coefficients shows that β 1 ∈ H 2 (Z 1 , ∂Z 1 ; R) is an integral class. Analogously, β 2 is integral. Thus, also the glued class α is integral.
Setup 6.13 (decomposable relators II). Let G 1 be a group with H 2 (G 1 ; R) ∼ = 0 and let r 1 , r 2 ∈ G 1 . We then consider the group where t is a fresh generator of t ∼ = Z.
Also here, there is a canonical homology class α ∈ H 2 (T (G 1 , r 1 , r 2 ); R), which is defined as follows: Let X 1 be a model of BG 1 and let be the cylinder space associated with r 1 and r 2 . Let β ∈ H 2 (Z, ∂Z; R) be the relative "fundamental" class as in Corollary 6.9. Glueing the two cylindrical ends of Z by an orientation reversing homeomorphism leads to a CW-complex P such that π 1 (P ) ∼ = T (G 1 , r 1 , r 2 ) =: G in the obvious way (the additional generator t corresponds to the loop {1} × ([0, 1] s∼s [0, 1]) in the looped cylinder. Let α ∈ H 2 (P ; R) be the class obtained by glueing β to itself via the cylinder. Then we define α ∈ H 2 (G; R) as the image of α under the classifying map P → BG (which is induced by the canonical map X 1 → P and the cylinder loop).
Theorem 6.14 (decomposable relators). Let G 1 be a group with H 2 (G 1 ; R) ∼ = 0 and let r 1 ∈ G 1 be an element of infinite order.
2. Let r 2 ∈ G 1 be an element of infinite order, let α ∈ H 2 (T (G 1 , r 1 , r 2 ); R) be the canonical class (Setup 6.13), and let t be the fresh letter in T (G 1 , r 1 , r 2 ). Then Proof. In both cases, we will use that the stable commutator length and the canonical CW-complexes of decomposable relators can be expressed in terms of the stable commutator lengths and cylinder complexes of the sub-relators. Ad 1. It is known that [Cal09a, Proposition 2.99] scl G1 * G2 (r 1 · r 2 ) = scl G1 r 1 + scl G2 r 2 + 1 2 .
We will now show that α 1 equals the right-hand side. In the following, we will use the notation from Setup 6.11. By construction, we have where c : P → BD(G 1 , G 2 , r 1 , r 2 ) is the classifying map of P . The mapping theorem (Corollary 3.5) shows that α 1 = α 1 . Therefore, it suffices to compute α 1 . Because r 1 and r 2 have infinite order, the inclusions of S 1 × {1} into Z 1 and Z 2 , respectively, are π 1 -injective. As π 1 (S 1 × {1}) ∼ = Z is amenable, the amenable glueing theorem [BBF + 14, Section 6] shows that α 1 = β 1 1 + β 2 1 ; the proofs of Bucher et al. carry over from the manifold case to this setting, because they established the necessary tools in bounded cohomology in this full generality. Moreover, we know that (Corollary 6.9) β 1 1 = 4 · scl G1 r 1 and β 2 1 = 4 · scl G2 r 2 .
We will now use the notation from Setup 6.13. The classifying map c : P → BT (G 1 , r 1 , r 2 ) maps α to the canonical class α ∈ H 2 (G; R) and the mapping theorem (Corollary 3.5) shows that α 1 = α 1 . Because r 1 and r 2 have infinite order, we can again use the amenable glueing theorem to deduce that α 1 = β 1 .

Simplicial volume as filling invariant
We mention that also simplicial volume of higher-dimensional manifolds admits a description as a filling invariant (this result will not be used in the rest of the paper): Proof. Ad 1. This is a special case of Proposition 6.8: We consider Z := M \∆ • . The map σ :

Proof of Theorems C and D
Theorem C is a special case of Theorem 6.14 with the decomposable relators of Setup 6.11. Let G be a group that satisfies H 2 (G; R) ∼ = 0 and let r ∈ G be an element of infinite order. Then we define the double D(G, r) of G and r by setting where G left and G right are isomorphic copies of G and r right ∈ G right , r left ∈ G left are the elements corresponding to r ∈ G. Observe that if G is finitely presented, then so is D(G, r). As in Setup 6.11 there is a canonical integral class α ∈ H 2 (D(G, r); R).
Corollary 6.16 (Theorem C). Let G be a group with H 2 (G; R) ∼ = 0 and let r ∈ G be of infinite order. Then the canonical integral class α ∈ H 2 (D(G, r); R) satisfies Proof. This is an immediate corollary of Theorem 6.14 (1).
Applying Theorem C to the universal central extension E of Thompson's group T , we deduce Theorem D: Corollary 6.17 (Theorem D). For every q ∈ Q ≥0 , there is a finitely presented group G q and an integral class α q ∈ H 2 (G q ; R) such that α q 1 = q. In particular, for every > 0 there is a finitely presented group G and an integral class α ∈ H 2 (G ; R) such that 0 < α ≤ .
Proof. Let q ∈ Q ≥0 . For q = 0 we can take the zero class of the trivial class (or any integral 2-class in any finitely presented amenable group).
For q > 0, let r q ∈ E be an element in the universal central extension E of Thompson's group T with scl E r q = q/8. Proposition 5.1 asserts that such an element exists, that E is finitely presented and that H 2 (E; Z) ∼ = 0 and hence H 2 (E; R) = 0. As scl E r q > 0 the element r q ∈ E has infinite order.
Let G q := D(E, r q ) be the double and let α q ∈ H 2 (G q ; R) be the associated integral 2-class. Theorem C shows that We note that one can prove the second part of Theorem D also via previously known examples of stable commutator length (Example 4.3.4).
7 The l 1 -semi-norm of products with surfaces: Proof of Theorem E Bucher [BK08] computed the simplicial volume of the product of two surfaces (see Theorem 2.10). We will use her techniques to generalise this statement to the product of more general 2-classes. This will allow us to construct integral 4-classes whose l 1 -semi-norm can be expressed in terms of the l 1 -semi-norm of 2-classes. Theorem E will be a corollary (Corollary 7.2) of these constructions.
Theorem 7.1. Let G and Γ be groups, let α ∈ H 2 (G; R), and β ∈ H 2 (Γ; R). Furthermore, let ρ : Γ → Homeo + (S 1 ) be a circle action of Γ. Assume furthermore that ρ * Eu ∈ C 2 b (Γ; R) is an extremal cocycle for β; see Section 3.6. Then the class α × β ∈ H 4 (G × Γ; R) satisfies Theorem 7.1 is a strict generalisation of Bucher's result [BK08] and our proof follows the outline of Bucher's work. Recall that for g ≥ 2 we denote the oriented closed connected surface of genus g by Σ g , its fundamental group by Γ g , and its fundamental class by [Σ g ] R ∈ H 2 (Σ g ; R) ∼ = H 2 (Γ g ; R). Fix a hyperbolic structure on Σ g and let ρ : Γ g → Homeo + (S 1 ) be the corresponding action on the boundary ∂Γ g ∼ = S 1 . The class ρ * Eu ∈ C 2 b (Γ g ; R) is extremal to [Σ g ] R (see Section 3.6) and satisfies Therefore, we obtain the following immediate corollary to Theorem 7.1: Corollary 7.2 (Theorem E). Let g ≥ 2, let G be a group, let α ∈ H 2 (G; R) and let Γ g and [Σ g ] R ∈ H 2 (Γ g ; R) be as above. Then the class α × Proof of Theorem 7.1. In this proof, all cocycles will be given in the homogeneous resolution. The upper bound holds for all classes in degree 2 (Corollary 2.11). For the lower bound we will use duality (Proposition 3.4): be the Euler class for the given action ρ : Γ → Homeo + (S 1 ). Moreover, let ω ∈ C 2 b (G; R) be an extremal cocycle for α ∈ H 2 (G; R) in the homogeneous resolution; see Proposition 3.4. By possibly replacing ω by alt 2 b (ω), we may assume that ω is alternating; see Section 3.5. By assumption, ω ∞ ≤ 1 and [ω], α = α 1 and Γ eu R b , β = β 1 .
• If more than three of the x i are identical or if exactly three of the x i are identical and the two remaining x i are identical, then the Or-term in Equation (2) always vanishes and we get that Θ((g 0 , γ 0 ), . . . , (g 4 , γ 4 )) = 0 < 2 3 .

Manufacturing manifolds with controlled simplicial volumes
The computation of 1 -semi-norms of 2-classes in group homology allows us to construct manifolds with controlled simplicial volume. This construction will involve a normed version of Thom's realisation theorem, which we recall in Section 8.1. Theorem A is proven in Section 8.2 and the theorems for dimension 4 (Theorems B and F) are proven in Section 8.3. Finally, in Section 8.4 we discuss related problems and further research topics.

Thom's realisation theorem
In order to turn classes in group homology into manifolds with controlled simplicial volume, we will use the following, normed, version of Thom's realisation theorem: Theorem 8.1 (normed Thom realisation). For each d ∈ N ≥4 , there exists a constant K d ∈ N >0 with the following property: If G is a finitely presented group (with model X of BG) and α ∈ H d (X; R) is an integral homology class, then there is an oriented closed connected d-manifold M , a continuous map f : M → X and a number m ∈ {1, . . . , K d } with Moreover, one can choose K 4 = 1 and K 5 = 1.
Proof. Everything except for the condition on the simplicial volume is contained in Thom's classical realisation theorems [NT07, Theorems III.3, III.4]. (Thom's original theorems apply to X because every singular homology class of X is supported on a finite subcomplex; as G is finitely presented, we can choose the subcomplex in such a way that the inclusion into X induces a π 1 -isomorphism.) One can then apply surgery to obtain a manifold representation of α, where f : M → X in addition is a π 1 -isomorphism [CL15, (proof of) Theorem 3.1] (this will not touch the multiplier m). Therefore, the mapping theorem for the l 1 -semi-norm (Corollary 3.5) shows that

No gaps in higher dimensions: Proof of Theorem A
We promote the computations of l 1 -semi-norms in degree 2 to higher dimensions using cross-products. The manifolds will then be provided by the normed Thom realisation (Theorem 8.1).
Proof of Theorem A. Let d ∈ N ≥4 . We fix an orientable closed connected hyperbolic (d − 2)-manifold N d ; in particular, N d > 0. Moreover, let K d ∈ N be the constant provided by Thom's realisation theorem (Theorem 8.1). Let ∈ R >0 . By Theorem D, there exists a finitely presented group G and an integral class α ∈ H 2 (G; R) with 0 < α 1 ≤ . Let X be a model of BG. Then the product class is integral and satisfies (by Proposition 2.9) The normed version of Thom's realisation theorem (Theorem 8.1) provides an orientable closed connected d-manifold M and a number m ∈ {1, . . . , K d } with M = m · α 1 .
We conclude that As the constants on the right hand side just depend on d, this shows that there is no gap at 0 in SV(d), the set of simplicial volumes of orientable closed connected d-manifolds. By additivity (Remark 2.3), the set SV(d) is also dense in R ≥0 .

Dimension 4: Proofs of Theorems B and F
In dimension 4, we have more control both on Thom realisation (Theorem 8.1) and on the l 1 -norm of integral 4-classes in group homology (Theorem E). This allows us to prove Theorems B and F.
Proposition 8.2. Let G be a finitely presented group and let α ∈ H 2 (G; R) be an integral class. Then there exists an orientable closed connected 4-manifold M α with M α = 6 · α 1 .
Proof. We proceed as in the proof of Theorem A and consider the product class of α with the fundamental class [Σ 2 ] R of a surface of genus 2. Observe that α is also integral. Then the normed Thom realisation (Theorem 8.1) shows that there exists an orientable closed connected 4-manifold M α with M α = α 1 . We now apply the norm computation from Corollary 7.2 and obtain Proof of Theorem B. We only need to combine Theorem D (which allows to realise any non-negative rational number as l 1 -semi-norm of an integral 2-class of a finitely presented group) with Proposition 8.2.
Moreover, we can summarise the relation between stable commutator length and simplicial volumes in dimension 4 as follows: Corollary 8.3 (Theorem F, dimension 4, exact values via scl). Let G be a finitely presented group with H 2 (G; R) ∼ = 0 and let g ∈ G be an element in the commutator subgroup. Then there exists an oriented closed connected 4manifold M g with M g = 48 · scl G g.
Proof. We may assume without loss of generality that r has infinite order (otherwise we can just take M = S 4 ). We again consider the doubled group D(G, r) (as in Corollary 6.16) and the canonical homology class α ∈ H 2 (D(G, r); R), which is integral (Remark 6.12); as G is finitely presented, also D(G, r) is finitely presented. Applying Corollary 6.16 shows that In combination with Proposition 8.2, we therefore obtain an orientable closed connected 4-manifold M with M = 6 · α 1 = 48 · scl G r.
Remark 8.4. The concrete example manifolds in the proof of Theorem B, in general, might have different fundamental group; however, by construction, their first Betti numbers are uniformly bounded: For each q ∈ Q, we have

Outlook on related problems
We are confident that Theorem D or Theorem F may also be applied to prove the existence of oriented closed connected 4-manifolds with (arbitrarily small) transcendental simplicial volume. We will explain this in more detail in Appendix A.3. Our techniques for manufacturing manifolds with controlled simplicial volumes are based on group-theoretic methods and not on genuine manifold-geometric constructions. One might wonder whether Theorem A also holds under additional topological or geometric conditions such as asphericity or curvature conditions.
Originally, we set out to study simplicial volume of one-relator groups and its relation with stable commutator length. However, we then realised that some of the techniques applied in a much broader context (with a weak homological condition). In a forthcoming article [HL], we will continue our investigations. We discuss a connection between the l 1 -semi-norm of the relator-class in onerelator group with the stable commutator length of the relator in the free group.

A Stein-Thompson groups
Euler central extensions of Stein-Thompson groups are a source of transcendental stable commutator lengths. In light of Theorem F they might hence be helpful for the construction of manifolds with (small) transcendental simplicial volume. In this appendix, we survey known results of stable commutator length on Euler central extensions of Stein-Thompson groups (Section A.1). We then discuss properties of universal central extension of Stein-Thompson groups (Section A.2). We believe that those groups also have transcendental stable commutator length. This would follow from Conjecture A.5 on the bounded cohomology of those Stein-Thompson groups. In Section A.3, we discuss potential applications to simplicial volumes.

A.1 Transcendental stable commutator length
It was a question by Gromov if scl is rational on finitely presented groups. Zhuang answered this question in the negative by constructing finitely presented groups with transcendental scl [Zhu08,Zhu09]. These groups are certain central extensions of Stein-Thompson groups. We briefly describe his construction.
In 1992, Stein generalised Thompson's groups T and F to allow for a wider set of breakpoints and slopes [Ste92][Cal09a, Section 5]. We consider a special case of this generalisation. Let p, q ∈ N be natural numbers and denote by p, q the multiplicative subgroup of R + generated by p and q. Then We call F p,q and T p,q Stein-Thompson groups (but one should note that the original definition of Stein is more general). This construction generalises the original Thompson groups via T = T 1,2 and F = F 1,2 . Stein showed that these groups are always finitely presented. The groups T p,q act on the circle via the inclusion T p,q → Homeo + (S 1 ). Let Tp,q eu Z ∈ H 2 (T p,q ; Z) be the corresponding Euler class (see Remark 3.7) and let T p,q be the corresponding central extension 1 / / Z / / T p,q / / T p,q / / 1, as described in Section 3.6.
Theorem A.1 ([Zhu08, Theorem 3.9]). Let p, q be integers that freely generate p, q and satisfy that gcd(p − 1, q − 1) = 1 and let T p,q be the central extension of T p,q by the Euler class Tp,q eu Z . Then there is a homogeneous quasimorphism rot : T p,q → R of defect 1, called rotation number, which generates the space of all homogeneous quasimorphisms on T p,q . Hence, scl Tp,q (f ) = rot(f ) 2 .
for all f ∈ T p,q .
The rotation number is well-studied and has geometric meaning [BFH16,Ghy87]. Liousse [Lio08] calculated the rotation numbers for T p,q : Theorem A.2 ([Lio08, Theorem 2.C'][Cal09a, Theorem 5.17]). Suppose that p, q freely generate p, q and suppose that gcd(p − 1, q − 1) = 1. Any number of the form log(α) log(β) mod Z where α, β ∈ p, q can be realised as the rotation number of an element in T p,q .
Passing to T p,q , we can also realise such values log(α) log(β) on the nose (and not only modulo 1).
The theorem of Gel'fond and Schneider [Gel69,Sch35] implies that this number is transcendental. The same arguments also show: For each n ∈ N >0 , there exists a g ∈ T 2,3 with scl T2,3 g = log (3) n·log(2) , which is still transcendental (and small).
These examples form a rich class of groups with interesting stable commutator length. However, they are not quite fit for our purposes. Recall that we are interested in groups with vanishing real second homology. Ghys and Sergiescu [GS87] showed that, e.g., H 2 ( T 1,2 ; R) ∼ = R.

A.2 The universal central extension
We will now explain a possible approach for generalising the previous examples to realising transcendental values under the additional homological condition that H 2 ( · ; R) ∼ = 0. Indeed, we believe that Proposition 5.6 may be generalised to Stein-Thompson groups: Definition A.4 (Euler-generated). We say that a subgroup Γ of PL + (S 1 ) is Euler-generated if the following conditions are satisfied: • H 2 b (Γ; R) is one-dimensional and generated by the Euler class Γ eu R b , • and the evaluation homomorphism H 2 (Γ; Z) → Z associated with the integral Euler class Γ eu Z is surjective.
Conjecture A.5. Let T p,q be a generalised Stein-Thompson group as in Section A.1. Then T p,q is Euler-generated.

A.2 The universal central extension
Remark A.6. If T p,q fails to satisfy that H 2 b (T p,q , R) ∼ = R, generated by the Euler class, then this would imply that F p,q is not amenable.
For suppose that F p,q is amenable; then H 2 b (T p,q ; R) may be computed using a resolution of bounded alternating functions on S p,q , where S p,q ⊂ S 1 is the set of points in S 1 which also lie in Z[ 1 pq ]. This is because T p,q acts in this case on S p,q with (amenable?) stabilisers of isomorphism type F p,q . Using moreover that T p,q acts 3-transitively on S p,q , we see that the "only" bounded cocycle is the Euler cocycle. (Amenable resolutions are one way to compute bounded cohomology [Fri17,Chapter 4.9].) While we do not make any conjectures towards the amenability of F p,q we believe that more subtle methods are needed to show a possible non-amenability of F p,q .
An affirmative answer to Conjecture A.5 could be used as input for the following analogue of Proposition 5.1: Because ϕ : E → Γ is surjective, scl E and scl Γ have the same image.
However, a priori it does not seem clear why it should also be surjective (it could be an integral multiple of another class in H 2 (Γ; Z)).
In any case, Thompson's group T is Euler-generated (Example A.10); hence, applying the mapping Lemma A.8 to T gives an alternative proof of the penultimate step of the proof of Proposition 5.1.
Example A.10. Thompson's group T is perfect and H 2 b (T ; R) ∼ = R, generated by the Euler class T eu R b (Proposition 5.6). Moreover, because the integral Euler class is part of a basis of H 2 (T ; Z) (Theorem 5.5), we obtain that the evaluation homomorphism H 2 (T ; Z) → Z associated with the integral Euler class of T is surjective.

A.3 Transcendental simplicial volumes of 4-manifolds
We will now sketch a possible approach to show the existence of orientable closed connected 4-manifolds with (arbitrarily small) transcendental simplicial volume, using Theorem F and Stein-Thompson groups. As a direct consequence of Theorem F we have: Corollary A.11 (dimension 4, inheritance of transcendence). If there exists a finitely presented group G with H 2 (G; R) ∼ = 0 and an element r ∈ G such that scl G r is transcendental, then there also exists an oriented closed connected 4-manifold M such that M = 48 · scl G r is transcendental.
In fact, the only remaining obstacle to apply Corollary A.11 is to ensure that the homological criterion H 2 ( · ; R) ∼ = 0 is satisfied. Natural candidates for such groups are universal central extensions of the Stein-Thompson groups.
Corollary A.12 (dimension 4, inheritance of transcendence). If Conjecture A.5 holds for the Stein-Thompson group T 2,3 , then for every ∈ R >0 there exists an orientable closed connected 4-manifold M such that M is transcendental and satisfies 0 < M < .
Proof. According to Proposition A.7, we could then use the universal central extension of T 2,3 as the input for Corollary A.11.
Remark A.13. More generally, we have (by Proposition 8.2): If there exists a finitely presented group G with an integral homology class α ∈ H 2 (G; R) such that α 1 is transcendental, then there also is an orientable closed connected 4-manifold M such that M is transcendental.