The spectrum of simplicial volume

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CORSO DI LAUREA IN MATEMATICA Laurea Magistrale in Matematica

The spectrum of simplicial volume

Tesi di Laurea Magistrale

CANDIDATO:

Giuseppe Bargagnati

RELATORE:

Prof. Roberto Frigerio

CONTRORELATORE:

Prof. Bruno Martelli

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Introduction

The simplicial volume is a homotopy invariant of compact manifolds introduced by Mikhail Gromov in his pionieristic article of 1982 ([15]), that associates to a manifold the `1-norm of its real fundamental class. Although this invariant has a purely topological interpretation (and is in fact a homotopy invariant), it was noticed that it is heavily related with other geometric invariants.

Until last year, very little was known about the structure of the set SV (d) of simplicial volumes of oriented closed connected d-manifolds for dimensions d greater than or equal to 4. In lower dimensions, this set is fully understood: in dimensions 0 and 1, it can be easily shown that SV (0) = {1} and SV (1) = {0}. For d = 2 the simplicial volume of an oriented closed connected surface is tied to the Euler characteristic.

Theorem 0.1 ([15]). Let g ≥ 1 be a natural number and let Σg be the oriented closed

connected surface of genus g. Then ||Σg|| = 2 · |χ(Σg)| = 4 · (g − 1), and we have that

SV (2) = {0, 4, 8, . . .} = N[4].

For d = 3 things are more complicated, and we have the following equality. Theorem 0.2 ([15], [12, Corollary 7.8]).

SV (3) = Nhvol(M ) v3

M complete hyperbolic three manifold with toroidal boundary and finite volumei, where v3 denotes the maximum volume of an ideal simplex in H3.

In both these dimensions, the set SV (d) presents a gap, i.e. there exists a constant Kd such that the simplicial volume of an oriented closed connected d-manifold either

vanishes or is greater than or equal to K_d. It was a long-standing open question whether this phenomenon occurs also in higher dimensions or it is exclusive of dimensions 2 and 3. In the article "The spectrum of simplicial volume" ([25]), which dates back to 2019, Nicolaus Heuer and Clara Löh answer this question in the negative sense. In this thesis, we carefully analyze the article of Heuer and Löh, giving the necessary backgroud to understand their proof of the density of SV (d) in R≥0 for d ≥ 4 and of the rational

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First, we introduce the basic toolkit for the development of the thesis: we give the precise definition of simplicial volume and state some of its basic properties, give a brief introduction to bounded cohomology of discrete groups, define the Euler class and the Orientation class and take a short survey on amenable and hyperbolic groups.

In the second chapter, we introduce commutator length and its stabilization, stable commutator length.

Definition 0.3. Let G be a group and let g ∈ G0 be an element of the commutator subgroup. The commutator length of g, denoted with cl(g), is defined as the least number of commutators whose product is g.

The stable commutator length of g, which is denoted by scl(g), is defined as the following limit

scl(g) = lim

n!∞

cl(gn) n .

Although these concepts are stated in a purely algebraic fashion, they can be rein-terpreted in a geometric way. After giving some basic properties of these algebraic objects, we state Bavard’s Duality Theorem ([1]), that is crucial in the computation of stable commutator length. Then, we introduce Thompson’s group T (which was the first example of an infinite finitely presented simple group) and its universal central extension E. The remaining part of the chapter is devoted to proving that the spectrum of stable commutator length on E is Q≥0.

In the third chapter we give the definition of filling norm and stable filling norm, and explore their connection with stable commutator length. This connection, and the results of Chapter 2 about stable commutator length on the group E, will be the key to prove the following theorem.

Theorem 0.4. Let G be a group with H2(G, R) = 0 and let r ∈ G0be an element of infinite

order. Then there exists a group D(G, r) and a 2-homology class α ∈ H₂_{(D(G, r), R)} such that

||α||1 = 8 · sclG(r).

Since the universal central extension of the Thompson’s group T satisfies H₂_{(E, R) = 0,} applying this theorem to E we are able to construct groups along with 2-homology classes with a prescibed positive rational `1-norm.

In the fourth chapter, these classes with prescribed norm are promoted to higher dimension classes taking cross products in homology; the norm of these products can be controlled thanks to some general estimates on the `1-norm of cross products.

Proposition 0.5. Let X, Y be topological spaces, let m, n ∈ N and let α ∈ Hm(X, R),

β ∈ Hn(Y, R). Then the cross product α × β ∈ Hm+n(X × Y, R) satisfies the following

inequalities:

||α|| · ||β|| ≤ ||α × β|| ≤ m + n m

!

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3 A refined version of a theorem by Thom ([28]) allow us to convert these homology classes with controlled `1-norm into oriented closed connected manifolds with controlled simplicial volume. Thus, we show the following theorem.

Theorem 0.6. Let d ≥ 4 be a natural number; then the set SV (d) is dense in R≥0.

In dimension 4, thanks to a generalization of a result of Michelle Bucher-Karlsson on the simplicial volume of products of surfaces ([4]), this result can be improved.

Theorem 0.7. The following containment holds:

Q≥0⊂ SV (4).

In a subsequent article ([26]), the authors have shown that this containment is strict. Finally, we devote an appendix to the proof of Bavard’s Duality Theorem, that is crucial in the computations of stable commutator length.

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Chapter 1

Preliminaries

In this chapter we give the definition of simplicial volume, along with some basic properties; we introduce also the machinery of bounded cohomology, which turns out to be a fundamental tool in the whole thesis. Then, we introduce the notion of amenability, which is closely related to these topics and has a very important role in the whole thesis. Eventually, we introduce another class of groups, namely hyperbolic groups. For a general reference, we refer to [12].

1.1 Simplicial volume: definition and basic properties

In order to define simplicial volume, we have to introduce an `1-norm on the complex of the singular chains with real coefficients Cn(X, R) of a topological space X. For every

n ≥ 0, let P

aiσi be a (finite) sum of singular n-simplices; we define the `1-norm

X aiσi ₁ = X |ai|

which descends to a seminorm k·k₁ on H_n(X).

If M is a closed (i.e. compact with empty boundary) connected oriented n-manifold, then H_n_{(M, Z) is isomorphic to Z; the orientation allows us to choose a preferred} generator, which is denoted by [M ]_Z and is called fundamental class. Now let us denote with [M ] = [M ]_R ∈ Hn(M, R) the image of [M ]Z under the change of coefficients map

Hn(M, Z) ! Hn(M, R). The simplicial volume of M is defined as

kM k = k[M ]k₁.

We notice that the change of the orientation turns [M ] into −[M ], but does not affect the `1-norm, since kP

aiσik =P|ai| = kP−aiσik; this means that the simplicial volume

does not depend on the orientation chosen for M , so it is a well defined invariant of orientable manifolds. If M is not orientable and we denote with M the orientablef connected double covering of M , we set kM k =

eM

2 . Finally, if M is compact and

disconnected we can define the simplicial volume as the sum of the simplicial volumes of 5

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the connected components of M .

Even if M is compact with non-empty boundary we can define a "relative" version of the simplicial volume for the pair (M, ∂M ). Let us identify C_n_{(∂M, R) with its} image in Cn(M, R) under the map induced by the natural inclusion ∂M ,! M ; we have

that Cn(∂M, R) is closed in Cn(M, R) with respect to the `1-norm. It follows that the

quotient seminorm on the space C_n_{(M, ∂M, R) ' C}_n_{(M, R)/C}_n_{(∂M, R) is indeed a norm.} Moreover, if M is connected and oriented, also Hn(M, ∂M, Z) is cyclic infinite with a

preferred generator. Keeping the same notations as above, we can define the simplicial volume of (M, ∂M ) as

kM, ∂M k = k[M, ∂M ]k₁.

The procedure to extend this definition to the case of not orientable and/or disconnected manifolds is word by word the same discussed above.

Let M and N be n-manifolds (possibly with non-empty boundary) and let f : (M, ∂M )! (N, ∂N ) be a map of pairs; we denote with H•(f ) the map induced in homology. We

have that H•(f )( X aiσi) = X ai(f ◦ σi),

where f ◦ σi is still a singular simplex in (N, ∂N ); this implies that the map induced by

f is a norm-non-increasing map, and so we can deduce the following inequalities k(N, ∂N )k = k[N, ∂N ]k₁ = kHn(f )([M, ∂M ])k1 |d| ≤ k[M, ∂M ]k₁ |d| = kM, ∂M k |d| . If kN, ∂N k is bounded, it turns out that there is a restriction on the possible degrees of maps of pairs between M and N . If M = N , and f is a map of degree d with |d| ≥ 2, iterating this map, by multiplicativity of the degree, we obtain maps with arbitrary large degree. As a consequence, we have the following corollary.

Corollary 1.1. If a manifold M admits a self-map of degree different from −1, 0, 1

then kM, ∂M k = 0. For example, Dn_{, S}n−1

= 0 if n ≥ 2 and kSnk = 0 if n ≥ 1. Another property of simplicial volume is the multiplicativity with respect to finite coverings.

Proposition 1.2. Let p : M ! N be a covering of degree d of manifolds (possibly with

boundary); then

kM, ∂M k = d · kN, ∂N k .

Proof. Every single simplex σ in N lifts to d single simplices_eσj (with j = 1, . . . , d) in N (since it is defined on a simply connected space); moreover, ifP

i∈Iaisi is a fundamental

cycle for N , thenPd

j=1

P

i∈Iaies

j

i is a fundamental cycle for M . This gives the following

inequality

kM, ∂M k ≤ d · kN, ∂N k .

We have already shown that the other inequality holds for maps of degree d, and since degree-d coverings are in particular maps of degree d, we get the conclusion.

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1.2. SIMPLICIAL VOLUME IN LOW DIMENSIONS 7

1.2 Simplicial volume in low dimensions

Let d be a natural number, and let us denote with SV (d) = {||M ||

M oriented closed connected d-manifold}

the spectrum of simplicial volume in dimension d. It is a natural question whether, given a positive real number c and a natural number d, c belongs to SV (d), i.e. if there exists an oriented closed connected d-manifold with simplicial volume equal to c. A first observation is that since simplicial volume is a homotopic invariant, as a consequence of the following theorem we have no chances of realizing every positive real number as simplicial volume of some manifold M .

Theorem 1.3 ([22]). There are countably many homotopy types of oriented closed

connected manifolds.

A characteristic of the set SV (d) is the invariance under addition; this is a consequence of explicit calculations (that we will show below) in dimensions 0, 1 and 2, and a direct consequence of the following theorem in dimension greater than or equal to 3.

Theorem 1.4 ([15], [12, Corollary 7.7]). Let d ≥ 3 be a natural number and let M, N be

oriented closed connected d-manifolds. Then

||M #N || = ||M || + ||N ||.

In low dimensions, we can describe explicitely the set SV (d). For d = 0, we have trivially that SV (0) = {1}, the only connected manifold being a single point. If d = 1, since the only closed connected 1-manifold is the circle S1, by Corollary 1.1 we have that SV (1) = {0}. Things get more complicated in dimension 2; we have the following theorem, which is due to Gromov.

Theorem 1.5 ([15]). Let g ≥ 1 be a natural number and let Σg be the oriented closed

connected surface of genus g. Then ||Σg|| = 2 · |χ(Σg)| = 4 · (g − 1), and so we have that

SV (2) = {0, 4, 8, . . .} = N[4].

We observe that in dimension 2 there is a gap, i.e. if a manifold has non-vanishing simplicial volume, then this has to be not smaller than a fixed positive number, which in this case is 4. This phenomenon occurs also in dimension 3, although the order type of the set SV (3) ⊂ R≥0 is different from that of SV (2).

Theorem 1.6 ([15], [12, Corollary 7.8]).

SV (3) = Nhvol(M ) v3

M complete hyperbolic three manifold with toroidal boundary and finite volumei, where v₃ _{denotes the maximum volume of an ideal simplex in H}3.

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This connection between simplicial volume and riemannian volume of hyperbolic manifolds is not exclusive of dimension three.

Theorem 1.7 ([15], [12, Theorem 7.4]). Let M be an oriented closed connected hyperbolic

manifold of dimension n. Then

||M || = V ol(M ) vn

,

where v_n _{is the maximum volume of an ideal simplex in H}n.

A result of Thurston ([29]) implies that SV (3) has the order type of ωω, and so has infinitely many accumulation points and presents a gap. By results of [11] we can estimate precisely this gap, which is given by w/v3 ≈ 0.928 . . ., where w is the volume of

the Weeks manifold.

Before the paper of Nicolaus Heuer and Clara Löh ([25]), we knew very little about the set SV (4); the smallest known simplicial volume of an hyperbolic oriented closed connected 4-manifold is equal to 64·π_3·v2

4 , where v4 is the maximal volume of an ideal simplex in H

4_.

In a paper published in 2008 ([4]), Michelle Bucher showed that the simplicial volume of the product of two oriented closed connected surfaces of genus respectively g and h is given by the formula ||Σg× Σh|| = 3₂ · ||Σg|| · ||Σh||; hence, ||Σ2× Σ2|| = 24. This has

been the smallest known simplicial volume of a 4-manifold until the paper of Heuer and Löh came out. Even the existence of a gap in dimension 4 was an open problem. Also the order type and the elements of the set SV (d) for d ≥ 5 were sorrounded by mistery.

1.3 Bounded cohomology of groups and spaces

The first mathematician who sistematically studied bounded cohomology of discrete groups was Gromov, in his pionieristic article of 1982 ([15]); although it is possible to present this topic in much greater generality, we will define it only with integer or real coefficients, and for us the action of the group on the coefficient ring will always be trivial.

We start by recalling the costruction and some properties of group cohomology. Let G be a group (that has to be always thought endowed with the discrete topology) and V be R or Z, considered as trivial (left) G-modules. For every n ∈ N we set

Cn(G, V ) := {f : Gn+1 ! V },

and we call it the module of the n-cochains; the G-module structure is given by (g · φ)(g0, . . . , gn) = φ(g−1g0, . . . , g−1gn). Define now the boundary maps δn: Cn(G, V )!

Cn+1_{(G, V ) as follows:} (δnf )(g0, . . . , gn+1) = n+1 X i=0 (−1)if (g0, . . . , ˆgi, . . . , gn+1).

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1.3. BOUNDED COHOMOLOGY OF GROUPS AND SPACES 9 One can easily check that δn+1◦ δn_{= 0 for every n ∈ N, and that δ}• _{is a G-map, i. e.}

that δ•(g · x) = g · δ•(x). So we have the following definition.

Definition 1.8. The pair (C•(G, V ), δ•) is a complex of (left) G-modules, called the homogeneous complex.

Now let us denote with Cn(G, V )G the submodule of the G-invariants of Cn(G, V ), i.e. the fixed points of the action of G. Since δnis a G-map, we have that (C•(G, V ), δ•)G forms a subcomplex of the homogeneous complex. Now, if we put

Zn(G, V ) = ker(δn) ∩ Cn(G, V )G and Bn(G, V ) = δn−1(Cn−1(G, V )G) with B0(G, V ) = 0, we have the obvious inclusion Bn(G, V ) ⊂ Zn(G, V ), and finally we set

Hn(G, V ) = Zn(G, V )/Bn(G, V ).

Definition 1.9. The V -module Hn(G, V ) is the n-th cohomology module of G with coefficients in V .

This definition could be presented in a more general setting, where V is an R[G]-module over some ring R (not necessarily R or Z), and also the action of G on V need not be trivial. However, for our purpose, it is sufficient to consider this specific case, in which things turn easier.

Remark 1.10. It follows from the definitions that H0(G, V ) = V. In fact (δ0(f ))(g0, g1) = f (g1) − f (g0),

so δ0f = 0 if and only if f is constant; on the other hand, constant maps are obviously G-invariant.

Remark 1.11 (Functoriality). Group cohomology is functorial both in the coefficients and in the group; functoriality is covariant in the former case and controvariant in the latter.

We can give a topological interpretation of group cohomology, but first we have to introduce a new object.

Definition 1.12. An Eilenberg-MacLane space of type K(G, 1) is a topological space

X with the following properties: • π1(X) = G;

• πn(X) = 0 for every n ≥ 2.

We may refer to an Eilenberg-MacLane space of type K(G, 1) also as a classifying space for the group G; we will usually denote it with the notation BG. The existence of such spaces is guaranteed by the following theorem.

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Theorem 1.13 ([17, Theorem 1.B.8]). For every group G there exists an

Eilenberg-MacLane space of type K(G, 1) which is a CW-complex. Moreover, as a consequence of the Whitehead theorem, this space is unique up to homotopy equivalence.

Now, let X be any model of a K(G, 1) and let us denote with C•(X, R) the module of the singular cochains of X with coefficients in the ring R; we can define the i-th group cohomology module of G by Hi_{(G, R) = H}i_{(X, R). It is a standard fact that this}

construction agrees with the one defined above; for further details, see [17], Example 1.B.7. In order to define bounded cohomology, we consider the `∞-norm on Cn(G; V ), that is

kf k_∞= sup{kf (g₀, . . . , gn)k |(g0, . . . , gn) ∈ Gn+1},

where V (which in our case is equal to R or Z) is endowed with the usual euclidean norm. We observe that, basically by definition, we have kg · f k_∞= kf k_∞ for every g ∈ G. Let us denote with Cn(G, V )_b the submodule of bounded n-cochains with respect to the `∞-norm; it is immediate to check that the differential carries C_bn(G, V ) to C_bn+1(G, V ). So if we set as before

Z_bn(G, V ) = ker(δn) ∩ C_bn(G, V )G and B_bn(G, V ) = δn−1(C_bn−1(G, V )G) where δ is the same differential defined before, we set

H_bn(G, V ) = Z_bn(G, V )/Bn_b(G, V ).

The norm on Z_bn(G, V ) descends to a seminorm on H_bn(G, V ) taking the infimum over all the representatives of a coclass; that is, if α ∈ H_bn(G, V ) we set

kαk_∞= inf{kf k_∞|f ∈ Zn

b(G, V ), [f ] = α}.

Definition 1.14. The V -module Hn

b(G, V ) is called n-th bounded cohomology module

of G with coefficients in V . The `∞-seminorm on H_bn(G, V ) is called the canonical seminorm.

Remark 1.15. The canonical seminorm is a norm if and only if Bn

b(G, V ) is a closed

subspace of Z_bn(G, V ).

Remark 1.16. Remark 1.11 applies also in the case of bounded group cohomology.

The natural inclusion C_b•(G, V ) ,! C•(G, V ) induces a map c•: H_b•(G, V )! H•(G, V )

that is called the comparison map. This map in general is neither injective nor surjective; its kernel is denoted by EH_b•(G, V ), and is called the exact bounded cohomology of G with coefficients in V .

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1.3. BOUNDED COHOMOLOGY OF GROUPS AND SPACES 11 Now, let X be a topological space and let us denote with Sn(X) the set of the

singular n-simplices of X, and with Cn(X, V ) the maps from Sn(X) to V . For an element

f ∈ Cn(X, V ) we set

kf k_∞= sup{|f (σ)|σ ∈ Sn(X)} ∈ [0, +∞]

and denote with C_bn(X, V ) the submodule of the bounded cocycles with respect to this norm. It is straightforward to observe that the restriction of the singular coboundary map makes (C_b•(X, V ), δ•) a cochain complex, whose cohomology is denoted by H_b•(X, V ) and is called the bounded cohomology of X with coefficients in V . Taking the infimum over all the representative of a class, the `∞-norm introduced above induces a seminorm on H_bn(X, V ). The bounded cohomology of spaces is also functorial both in the space and in the coefficients; the relation between the bounded cohomology of spaces and groups is stated by the following theorem.

Theorem 1.17 ([15], [12, Theorem 5.5]). Let G be a group and let X be a model of

the classifying space BG. Then H_b•(G, V ) is canonically isometrically isomorphic to H_b•(X, V ).

It is worth noting, as pointed out in the next theorem, that the whole information about the bounded cohomology of a group G is captured by the bounded cohomology of a space that has G as fundamental group, totally ignoring the higher homotopy groups. Theorem 1.18 ([15], [19]; [12, Theorem 5.8]). Let X be a path-connected topological

space. Then H_b•(X, V ) is canonically isometrically isomorphic to H_b•(π1(X), V ).

1.3.1 Duality

Bounded cohomology reveals to be a useful tool to compute the `1-norm of homology classes. Let us denote with h , i : H_bn_{(X, R) × H}_n_{(X, R) ! R the map induced by the} evaluation of cochains on chains. We have the following result.

Theorem 1.19 (Duality principle; [12, Lemma 6.1]). Let X be a path-connected

topolog-ical space and let α ∈ Hn(X; R). Then

kαk₁= sup{hβ, αi| β ∈ H_bn_{(X; R), kβk}_∞≤ 1}. Moreover, the supremum is achieved.

The cocycles which realize the supremum and whose `∞-norm is equal to 1 are called extremal cocycles for α. Let us denote with α∗ the class of H_bn_{(X, R) such that} hα∗_{, αi = 1.}

Proposition 1.20 ([12, Proposition 7.10]). Let X be a path connected topological space

and let α ∈ H_n_{(X, R). Then} ||α|| =

(

0 if ||α∗||∞= +∞;

||α∗_||−1

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1.3.2 Inhomogeneous complex

We introduce here a slightly different approach to group cohomology, which turns to be useful in many situations.

The key observation is that an element f ∈ Cn(G, V )G is fully determined by the values it takes on the (n + 1)-tuples with 1 as first entry. So we can set

¯

C0(G, V ) = V, C¯n(G, V ) = Cn−1(G, V ) = {f : Gn! V }

and consider ¯Cn(G, V ) as a V -module for every n ∈ N. It is immediate to check that we have V -isomorphisms

Cn(G, V )G! ¯Cn(G, V )

φ!(g₁, . . . , gn)7! φ(1, g1, g1g2, . . . , g1· · · gn)

. The differential δ• turns into a new differential ¯δ• defined as follows:

¯

δ0(v)(g) = g · v − v = 0 v ∈ V = ¯C0(G, V ), g ∈ G, where we recall that V is a trivial G-module, and for n ≥ 1 we have

¯ δn(f )(g₁, . . . , gn+1) =f (g2, . . . , gn+1) + n X i=1 (−1)if (g1, . . . , gigi+1, . . . gn+1) + (−1)n+1f (g1, . . . , gn).

The complex ( ¯C•, ¯δ•) is known as bar resolution or inhomogeneous complex associated to the pair (G, V ). It follows from the construction and the isomorphisms described above that the cohomology of this complex is canonically isomorphic to H•(G, V ). Just as we did before, also bounded cohomology can be defined using this complex.

1.3.3 Alternating cochains

Let G be a group and let α ∈ C_bn(G, V ) be a bounded cochain. We say that α is alternating if, for every permutation σ ∈ S_n+1 we have that

α(g_σ(0), . . . , g_σ(n)) = sign(σ) · α(g₀, . . . , gn).

We can easily associate to every cochain α ∈ C_bn(G, V ) an alternating cochain, denoted by altn(α) and defined as follows:

altn(α)(g0, . . . , gn) = 1 (n + 1)! · X σ∈Sn+1 sign(σ) · α(g_σ(0), . . . , gσ(n)).

We observe that || altn(α)||∞≤ ||α||∞.

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1.4. BOUNDED COHOMOLOGY IN LOW DEGREES 13 the differential ∂nto Cb,altn defines a map with values in Cb,altn+1which satisfies ∂n+1◦∂n= 0,

so alternating cochains form a subcomplex C_b,alt• (G, V ). An important fact is that we can compute the bounded cohomology with real coefficients through the alternating complex. Theorem 1.21 ([12, Proposition 4.26]). Let G be a group. The cohomology of the complex

C_b,alt• (G, R) is isometrically isomorphic to the bounded cohomology with real coefficients of the complex C_b•_{(G, R). Moreover, for every cocycle α ∈ C}_bn_{(G, R) the alternating cocycle} associated altn(α) represents the same cohomology class as α in H_bn_{(G, R).}

1.4 Bounded cohomology in low degrees

In the case with trivial real or integer coefficients, it is possible to do some direct computations of the cohomology modules of a group in degrees lower than or equal to two.

1.4.1 Degrees zero and one

We have already seen (Remark 1.10) that H0_{(G, V ) = V ; the same computations show}

that also H_b0(G, V ) = V . Using the inhomogeneous resolution we can easily compute the first cohomology module with trivial real or integer coefficients. Let f be an element of

¯

C1_{(G, V ); since the action of G on V is trivial, we have that}

¯

δ1(f )(g1, g2) = f (g1) − f (g1g2) + f (g2),

so the kernel of ¯δ1 is equal to Hom(G, V ). Since ¯δ0 = 0 we have that H1(G, V ) = Hom(G, V ). If we take a look to the first bounded cohomology module, the same computations show that it is equal to the module of bounded homomorphisms with real or integer values; since a bounded homomorphism from G to R or Z is trivial we have that H_b1(G, V ) = 0.

1.4.2 Degree two

Let G and V be as above. A central extension of G by V is an exact sequence 0! V ! Gi 0 π! G ! 1

with the property that i(V ) ⊂ Z(G0), where Z(G0) denotes the center of G0. We say that two central extensions are equivalent if they fit in a commutative diagram as follows:

0 V G0 G 1 0 V Gf0 G 1. id i f π id ei eπ

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We notice that, by commutativity of the diagram, the Five lemma implies that f is an isomorphism.

We can associate to an exact sequence

0! V ! Gi 0 π! G ! 0

a cocycle φ ∈ ¯C2(G, V ) via the following procedure: let s : G! G0 be a map such that π ◦ s = idG; we put

φ(g1, g2) = s(g1g2)(−1)s(g1)s(g2).

It is easy to check that the image of φ is contained in the kernel of π; by exactness this is equal to the image of V under the map i, which is injective, so φ is indeed an element of ¯C2(G, V ). A straightforward calculation shows that φ may be identified with a cocycle, so it defines a class α ∈ H2(G, V ); moreover, the cocycles that arise from different choices of the map s differ by a coboundary. So every central extension C defines a class e(C) ∈ H2(G, V ). This procedure can also be inverted, as stated in the following theorem.

Theorem 1.22 ([12, Proposition 2.5]). There is a natural bijection between the group

H2(G, V ) and the equivalence classes of central extensions of G by V .

1.4.3 Quasimorphisms

In order to understand the second bounded cohomology group of G, we can reduce ourselves to study the kernel and the image of the comparison map

c2 : H_b2(G, V )! H2(G, V ). Let us start with a definition.

Definition 1.23. A quasimorphism is a map f : G! V such that ther exists a constant

D ≥ 0 that satisfies the following inequality for every g1, g2 ∈ G:

|f (g₁) + f (g₂) − f (g₁g2)| ≤ D.

The least D ≥ 0 for which the above inequality holds is called the defect of f . The space of quasimorphisms is a V -module, denoted by Q(G, V ).

In other words, we can say that a quasimorphism is an element of ¯C1(G, V ) which has bounded differential. Obviously, bounded maps (i.e. elements of ¯C_b1(G, V )) and ho-momorphisms (elements of ¯Z1(G, V ) = Hom(G, V )) are both quasimorphisms; moreover we notice that the intersection ¯C_b1(G, V ) ∩ Hom(G, V ) contains only the zero map. The space ¯C_b1(G, V ) ⊕ Hom(G, V ) descibes the "trivial" quasimorphisms on G.

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1.4. BOUNDED COHOMOLOGY IN LOW DEGREES 15 Proposition 1.24. There is an exact sequence

0! ¯C_b1(G, V ) ⊕ Hom(G, V ) ,! Q(G, V )! EHψ _b2(G, V )! 0,

where the map ψ is induced by the restriction of the differential ¯δ1 to Q(G, V ). In particular

Q(G, V )/C¯_b1(G, V ) ⊕ Hom(G, V )' EH_b2(G, V ).

Proof. Obviously the image of the first inclusion is contained in the kernel of ψ, since ψ is the restriction of the differential. On the other hand, let f ∈ Ker(ψ); then f belongs to the image of the differential of the complex of bounded cochains, which means exactly that it is the image of a bounded 1-cochain up to an element of the kernel of ¯δ1, i.e. a homomorphism.

The map ψ maps Q(G, V ) to the kernel of the comparison map: in fact by definition the differential of a quasimorphism is bounded and is a representative of the zero class in H2(G, V ) since it is a coboundary. Also the surjectivity of ψ follows directly from the definitions.

This proposition gives also a way to show that H_b2(G, V ) is non trivial, that is to find quasimorphisms that stay infinitely distant from any homomorphism.

1.4.4 Homogeneous quasimorphisms

Let us start with a new definition.

Definition 1.25. A quasimorphism f : G! V is homogeneous if

f (gn) = n · f (g) _{∀g ∈ G, ∀n ∈ Z.}

The space of homogeneous quasimorphisms is a submodule of Q(G, V ) and is denoted by Qh(G, V ).

It is clear that a bounded homogeneous quasimorphism has to be identically zero. Homogeneous quasimorphisms are quite important actors when bounded cohomology with real (trivial) coefficients is on the stage, as explained in the following proposition. Proposition 1.26. Let f ∈ Q(G, R) be a quasimorphism. Then, there exists a unique

homogeneous quasimorphism ¯f which stays at finite distance from f . Moreover, the following inequality holds:

f − ¯f _∞≤ D(f ).

Proof. Applying iteratively the definition of quasimorphism, we have that for every g ∈ G, m, n ∈ N

|f (gmn) − nf (gm)| ≤ (n − 1)D(f ). (1.1) Hence, applying the triangular inequality

f (gn) n − f (gm) m ≤ f (gn) n − f (gmn) mn + f (gmn) mn − f (gm) m ≤ 1 n+ 1 m D(f ).

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This tells us that the sequence f (g_nn) is a Cauchy sequence; the same result holds also for

f (g−n₎

(−n) . From the fact that f (g

n_{) + f (g}−n_{) ≤ f (1) + D(f ), dividing by n we can deduce}

that the limits

lim n!+∞ f (gn) n =n!−∞lim f (gn) n = ¯f (g)

exist and coincide for every g ∈ G. The inequality (1.1) implies also that f (g) − f (gn₎ n ≤ D(f ), so by passing to the limit we get the claimed inequality

¯ f − f

_∞≤ D(f ). This implies that ¯f stays at finite distance from f (in particular, ¯f itself is a quasimorphism). We have only to check that ¯f is homogeneous:

m · ¯f (g) = lim n!∞m · f (gn) n = limn!∞m · f (gmn) mn = limn!∞ f (gmn) n = ¯f (g m_).

From the proof of Proposition 1.26 it immediately follows that D( ¯f ) ≤ 4D(f ); it actually holds a better estimate, namely D( ¯f ) ≤ 2D(f ) (for a proof of this fact we refer to [7], Lemma 2.58).

As an immediate consequence of Propositions 1.24 and 1.26 we have the following corollary.

Corollary 1.27. The space of real quasimorphisms Q(G, R) decomposes as follows

Q(G, R) = Qh(G, R) ⊕ ¯C_b1(G, R).

The restriction of the differential to the space of homogeneous quasimorphisms induces an isomorphism

Qh(G, R)/Hom(G, R) ' EHb2(G, R).

Now let us consider the case of quasimorphisms on an abelian group Γ: we have that for every g₁, g2 ∈ Γ

|nf (g₁g2) − nf (g1) − nf (g2)| = |f (g1g2)n

− f (gn₁) − f (gn₂)| = |f (g₁ngn₂) − f (g₁n) − f (gn₂)| ≤ D(f ).

Dividing by n and passing to the limit for n ! ∞ we have the equality f(g₁g2) =

f (g1) + f (g2), i.e. f is a homomorphism. We have proved the following proposition.

Proposition 1.28. Every homogeneous quasimorphism on an abelian group is a

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1.5. EULER CLASS 17 Now, let G be a group, and let us denote with Z(G) the center of G. If f is a quasimorphism on G, for every z ∈ Z(G) and for every g ∈ G, doing exactly the same computations done above, we have that

|f (zg) − f (z) − f (g)| ≤ D(f )

n ∀n ∈ N.

Once again, passing to the limit for n! ∞ we get the following result.

Corollary 1.29. Let G be a group and let z ∈ Z(G) be an element of the center of G.

Then, for every homogeneous quasimorphism f on G and for every g ∈ G we have f (zg) = f (z) + f (g).

1.5 Euler class

We define the Euler class associated to a circle action, which will turn out to be useful in the following chapters. Let us define Homeo+(S1) as the group of orientation preserving homeomorphisms of the circle, which we identify with R/Z, the quotient of the real line by the subgroup of integers; let us denote by p : R ! S1 _{the universal covering}

map and with ^Homeo+(S1_{) the homeomorphisms of R obtained by lifting elements} of Homeo+_(S1_{). Since p is a universal covering, every f ∈ Homeo}+_(S1_{) lifts to a}

map f ∈ ^e Homeo

+

(S1), and every map f ∈ ^e Homeo

+

(S1) descends, via p, to a map p∗(f ) ∈ Homeoe +(S1) such that ]p_∗(f ) ∈ ^e Homeo

+

(S1). So, we have a surjective group homomorphism p∗ : Homeo^

+

(S1) ! Homeo+(S1) whose kernel is the subgroup of integral translations, which is isomorphic to Z and is central in ^Homeo+(S1). Hence, we have a central extension

0! Z! ^ι Homeo+(S1)! Homeop∗ +(S1)! 1 (•),

where ι(n) = τn, where τndenotes the translation by n. Thanks to Theorem 1.22 we can

associate to this extension a 2-cohomology class.

Definition 1.30. The coclass eu ∈ H2_(Homeo+_(S1_{), Z) associated to the central}

exten-sion (•) is called the Euler class of Homeo+(S1).

Let us fix x₀ _{∈ R. We can define a section of p}∗, namely sx0 : Homeo

+_(S1₎ _!

^

Homeo+(S1) by setting sx0(f ) =fgx0, where withgfx0 we denote the unique lifting of f

such that fg_x₀ − x₀ ∈ [0, 1). As in Subsection 1.4.2, we can use this section to define a cocycle cx0 ∈ C

2_(Homeo+_(S1_{), Z), which by construction represents the Euler class:}

cx0 : Homeo +_(S1_{) × Homeo}+_(S1₎_{! Z,} _{(f, g)}_{! ^}_{(f ◦ g)} x0 −1 g fx0ggx0 ∈ Z,

where we identify Z with its image ι(Z) ⊂ ^Homeo+(S1). It turns out that this cocycle is bounded and that if we change basepoint the (bounded) cohomology class does not change.

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Lemma 1.31 ([12, Lemma 10.3]). The cocycle cx0 takes values into the set {0, 1}.

Moreover, if we choose two different "basepoints" x₀, x1 ∈ R, then the difference of the

resulting cocycles c_x₀− c_x₁ is the coboundary of a bounded integral 1-cochain.

Definition 1.32. Let us fix x0 ∈ R, and let cx0 be the bounded cocycle introduced

above. Then, the bounded Euler class eu_b ∈ H2

b(Homeo+(S1), Z) is defined by setting

eu_b = [c_x₀].

Thanks to the previous lemma, the class eu_b is well defined; moreover, by construction, the comparison map H_b2(Homeo+(S1_{), Z) ! H}2(Homeo+(S1_{), Z) sends eu}b to eu.

From now on, we will denote with the upper case letters (Eu) cocycles which represent the Euler class, and with the lower case letters (eu) the corresponding classes. In particular, if we set G = Homeo+(S1), then euZ

b ∈ Hb2(G, Z), euRb ∈ Hb2(G, R), euZ∈ H2(G, Z) and

euR_{∈ H}2_{(G, R) denote the classes represented by Eu in the corresponding cohomology}

groups. If Γ is a group with a circle action ρ : Γ! Homeo+_(S1_{), then ρ}∗_euZ_:=Γ _euZ_∈

H2(Γ, Z) is the Euler class associated to the action ρ. As we have seen in Proposition 1.22, there is a central extension associated to this cohomology class:

0! Z !Γe ! Γ ! 1.

This extension is denoted by Γ and is called the Euler extension. It follows from thee proof of Theorem 1.22 that the group Γ can be explicitely described as the set Z × Γe with the operation (z, g)(z0, g0) = (z + z0+ ρ∗EuZ_{(1, g, g · g}0_{), g · g}0_).

Now, let x₁, x2, x3 be a triplet of ordered points on the circle S1. We define the

homo-geneous (bounded) cocycle Or, which represents the orientation of the triplet, in the following way: Or(x1, x2, x3) =       

1 if the triplet is oriented counterclockwise; −1 if the triplet is oriented clockwise;

0 if at least two of the three points coincide.

Checking that Or satisfies the (bounded) cocycle condition (i.e. its differential is zero) is an easy exercise.

We observe that the group G = Homeo+(S1) preserves Or; we can use this fact to induce a 2-cocycle on Homeo+(S1). Fix ζ ∈ S1; we define the orientation cocycle, that we will denote again by Or ∈ C2_(Homeo+_(S1_{), Z), via Or(g}

1, g2, g3) = Or(g1ζ, g2ζ, g3ζ). This

cocycle induce a cohomology class [Or] ∈ H_b2(Homeo+(S1_{), Z) which turns out to be} independent of the "basepoint". The following relation holds.

Lemma 1.33 ([6]). We have the following equality:

−2 · euZ

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1.6. AMENABLE GROUPS 19

1.6 Amenable groups

Amenable groups were introduced by Von Neumann in the study of the Banach-Tarski paradox; they reveal to be very important in the context of bounded cohomology and simplicial volume. In this section, we will give a definition of amenable groups (in the context of discrete groups) and we will state a few theorems that collocate certain classes of groups under the label of amenability. Eventually, we will state a result that shows how amenability can affect bounded cohomology. Let us start with a definition.

Definition 1.34. Let Γ be a group, and let us denote with `∞(Γ) = C_b0_{(Γ, R) the space of} bounded real continuous functions on Γ, endowed with the usual action g · f (x) = f (g−1x). A mean on Γ is a map m : `∞(Γ)! R satisfying the following properties:

(1) m is linear;

(2) if we denote by 1Γ the constant map 1Γ(g) = 1, then m(1Γ) = 1;

(3) m(f ) ≥ 0 for every non-negative f ∈ `∞(Γ). Moreover, if m satisfies

(4) m(g · f ) = m(f ) for every g ∈ Γ

then m is said to be left-invariant (or simply invariant). We are now ready to define amenable groups.

Definition 1.35. A group Γ is amenable if it admits a left-invariant mean.

Now we give a short list of equivalent conditions for amenability.

Lemma 1.36 ([12, Lemma 3.2]). The following conditions for a group Γ are equivalent:

(1) Γ is amenable;

(2) there exists a non-trivial left-invariant continuous functional φ ∈ `∞(Γ)0; (3) Γ admits a left-invariant finitely additive probability measure.

From Lemma 1.36, it follows immediately that finite groups are amenable (it suffices to consider the counting-points measure). It turned out very soon that another familiar class of groups is included in the amenable family. The following result is due to von Neumann.

Theorem 1.37 ([24]). Abelian groups are amenable.

Thanks to the following proposition, we can build a lot of amenable groups. Proposition 1.38. Let Γ and H be amenable groups. Then:

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(2) If there is an exact sequence

1! H ! Γ0 ! Γ ! 1, then Γ0 is amenable.

(3) Every direct union of amenable groups is amenable.

We call elementary amenable groups the smallest class of groups which contains finite and abelian groups and is closed with respect to direct union, forming quotients, forming extensions and taking subgroups. It was shown by Chuo Ching ([9]) that not all the amenable groups are elementary amenable. But which groups are not amenable? Proposition 1.39 ([20]). Suppose that Γ contains a free subgroup of rank greater than

or equal to 2. Then Γ is not amenable.

It was a long-standing open question whether Proposition 1.39 was also a necessary condition for a group not to be amenable. However, the answer is negative, and was given first by Ol’shanskii ([27]).

If a group is amenable, then all the real bounded cohomology modules vanish, as pointed out by the following proposition.

Proposition 1.40 ([12, Corollary 3.9]). Let Γ be an amenable group. Then Hn

b(Γ, R) = 0

for all n ≥ 1.

Proof. We want to provide a homotopy j• between the identity and the zero map of the complex C_b•_{(Γ, R).}

Let m be an invariant mean on Γ and let f ∈ C_bn+1_{(Γ, R). For every (g}0, . . . , gn) ∈ Γn+1

we consider the function

f : Γ! R, f (g) = f (g, g0, . . . , gn).

Obviously, f ∈ `∞(Γ), so we can set (jn+1(f ))(g0, . . . , gn) = m(f ). The map jn+1 :

C_bn+1(Γ, R) ! Cbn(Γ, R) is bounded, and is Γ-equivariant by Γ-invariance of the mean m.

The collection of the maps jn+1_{, n ∈ N provides a Γ-equivariant homotopy between the} identity and the zero map of C_bn_{(Γ, R) for n ≥ 1.}

This proof can be easily adapted to show that H_bn(Γ, V ) = 0 for every dual normed R[Γ]-module V , i.e. every V which is isomorphic (as normed R[Γ]-module) to the topological dual of some normed R[Γ]-module W . However, we will not need this generality in the thesis.

Now we state the mapping theorem, which makes precise how bounded cohomology is related to amenability.

Theorem 1.41 ([19]; [12, Corollary 5.11]). Let f : X ! Y be a continuous map between

path-connected topological spaces. If the induced map between the fundamental groups is surjective with amenable kernel, then H_b•_{(f, R) : H}_b•_{(Y, R) ! H}_b•_{(X, R) is an isometric} isomorphism.

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1.7. HYPERBOLIC GROUPS 21

1.7 Hyperbolic groups

Another important class of groups is that of hyperbolic groups, which were introduced by Gromov in a very influential paper which dates back to 1987 ([14]). In order to give an accurate definition of these groups, we need some preliminary notions. Let us start with the one of Cayley graph.

Definition 1.42. Let G be a group and let S be a generating set for G. The Cayley graph Cay(G, S) is a graph whose:

• set of vertices is G; • set of edges is

_{{g, g · s}|s ∈ S ∪ S}−1_{\ {e}} .

So, two vertices in a Cayley graph are adiacent if and only if they are connected by right multiplication by an element (or an inverse of an element) of the generating set S. Equipped with the graph metric (which associates to any couple of vertices the least number of edges needed to join them) Cay(G, S) is a metric space. Of course, we would like to find an appropriate notion of equivalence in the category of metric spaces for which the choice of two different generating sets S and S0 gives two Cayley graphs which are equivalent as metric spaces. In order to do that, we introduce the notion of quasi-isometry.

Definition 1.43. Let (X, dX) and (Y, dY) be metric spaces and let f : X! Y .

• The map f is a quasi-isometric embedding if there exist two positive real constants c and b such that f is a (c, b)-quasi isometric embedding, which means that for every x, x0 ∈ X the following inequalities hold:

1 c · dX x, x 0 − b ≤ d_Y f (x), f (x0) ≤ c · d_X x, x0 + b.

• A map f0: X ! Y has finite distance from f if there exists a positive constant c such that

dY f (x), f0(x)≤ c ∀x ∈ X.

• The map f is a quasi-isometry if it is a quasi-isometric embedding for which there exists a map g : Y ! X which is a quasi-inverse quasi-isometric embeddings, i.e. g is a quasi-isometric embedding and f ◦ g has finite distance from idY and g ◦ f has

finite distance from idX.

• The metric spaces (X, d_X) and (Y, d_Y) are quasi-isometric if there is a quasi-isometry between them, and in this case we write X ∼QI Y .

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Theorem 1.44 ([21, Proposition 5.2.5]). Let G be a finitely generated group and let S

and S0 be two generating sets for G. Then the map id_G : Cay(G, S)! Cay(G, S0) is a quasi-isometry. Hence, any two different Cayley graphs for a group G are quasi-isometric. There are several ways to define hyperbolic metric spaces; we choose the approach via triangles, since it is more geometric and very intuitive.

Definition 1.45. A metric space is geodesic if any two points x, y ∈ X are the end points of a geodesic segment, which we denote with [x, y].

Remark 1.46. The Cayley graph of a group G is a geodesic metric space.

Now, let us fix δ > 0, let X be a geodesic metric space and let x, y, z ∈ X. A geodesic triangle with vertices x, y and z is the union of the geodesic segments [x, y], [y, z] and [x, z]. If for any point m ∈ [x, y] there exists a point in [y, z] ∪ [z, x] that is distant less than δ from m (and the same happens if m belongs to the other edges of the triangle), then the triangle is said to be δ-slim.

Definition 1.47. Let δ > 0; a geodesic metric space X is δ-hyperbolic if every geodesic triangle in X is δ-slim. A metric space is Gromov-hyperbolic (or just hyperbolic) if it is δ-hyperbolic for some δ > 0.

The property of being hyperbolic is invariant under quasi-isometry.

Proposition 1.48 ([16, Chapter 5, Proposition 15]). Let (Y, dY) be a geodesic metric

space. If Y is quasi-isometric to a δ-hyperbolic space X, then there exists a δ0 > 0 such that Y is δ0-hyperbolic.

We are ready to define hyperbolic groups.

Definition 1.49. Let G be a finitely generated group and let S be a generating set for

G; let (X, dX) be the metric space obtained endowing Cay(G, S) with the graph metric.

We say that the group G is hyperbolic if (X, d_X) is a hyperbolic metric space.

By Theorem 1.44, the quasi-isometry type of X does not depend on the choice of the generating set S. Moreover, by Proposition 1.48, the property of being hyperbolic is invariant under quasi-isometry, so we can test Definition 1.49 on any Cayley graph of G (and so it is a property of the group itself).

Now we want to generalize the concept of boundary of the "canonical" hyperbolic space Hn to a generic δ-hyperbolic space. So, let X be a geodesic δ-hyperbolic space and let us fix a point O ∈ X, that will serve as "origin". A geodesic ray is a path given by an isometry γ : [0, +∞)! X such that γ(0) = O and such that the segment γ([0, t]) is a geodesic for every t ≥ 0. Two geodesic rays γ1, γ2 are equivalent if there exists a constant

K ≥ 0 such that d(γ1(t), γ2(t)) ≤ K for every t ∈ R; the equivalence class of a geodesic

ray γ is denoted by [γ].

Definition 1.50. The Gromov boundary of a geodesic hyperbolic space X is the set

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1.7. HYPERBOLIC GROUPS 23 We can topologize ∂X in the following way: let p ∈ ∂X and r > 0. We define the Gromov product of three points x, y, z in a metric space as (x, y)z := 1/2 · (d(x, z) +

d(y, z) − d(x, y)); geometrically, it measures how geodesics from z to x and y stay close before diverging. Now we define V (p, r) = {[q] ∈ ∂X|there exist geodesic rays [γ1] =

p, [γ2] = q, lim infs,t!∞(γ1(s), γ2(t))O ≥ r}; varying r, these {V (p, r)|r > 0} form a

fundamental system of neighborhoods of p, and define a topology on ∂X. Endowed with this topology, ∂X is a compact metric space. The Gromov boundary is a quasi-isometry invariant.

Proposition 1.51 ([16]). If two hyperbolic geodesic metric spaces are quasi-isometric,

then the quasi-isometry induces a homoemorphism between their boundaries.

The actions of a group on geodesic metric spaces can tell a lot of things about the nature of the group, especially in the context of hyperbolic groups. To make this statement more precise, let us start with a definition.

Definition 1.52. Let X be a geodesic proper metric space (i.e. X is a geodesic met-ric space whose closed balls are compact). Then a finitely generated group G acts geometrically on X if:

• each g ∈ G acts as an isometry of X;

• the action is cocompact (i.e. X/G is compact); • the action is properly discontinuous.

It turns out that the property of being hyperbolic for a group G is equivalent to that of admitting a geometric action on an adequate space.

Theorem 1.53 ([14]). Let G be a group. Then G is hyperbolic if and only if there exists

a hyperbolic proper geodesic metric space X on which G acts geometrically. In this case, we have a homeomorphism between their boundaries, i.e. ∂G ' ∂X.

Now, let g ≥ 2 and let Σg be an oriented closed connected surface of genus g. It

is well known that Σ_g admits a hyperbolic structure, and that it is diffeomorphic to a quotient of the hyperbolic plane H2/Γg, where Γg is a discrete subgroup of Isom+(H2)

without elliptic elements. Since Γg is the fundamental group of Σg, and since ∂H2 ' S1

(it can be easily seen for example in the Poincaré disc model) and H2 is a proper geodesic metric space, thanks to Theorem 1.53 we have the following corollary.

Corollary 1.54. Let g ≥ 2 be a natural number and let Σg be an oriented closed connected

surface of genus g. Let us denote with Γ_g its fundamental group. Then Γ_g is a hyperbolic group, and ∂Γg' S1.

We observe that a group acts by isometries on its Cayley graph, and this action is geometric. This action induces also an action of G on its boundary; in the case of fundamental groups of hyperbolic surfaces, this is a circle action, i.e. a homomorphism π1(Σg)! Homeo+(S1).

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Chapter 2

Stable commutator length in

Thompson’s group T

In this chapter we introduce the tool of stable commutator length, which will turn to be very useful to compute the `1-norm of some cohomology classes later on; in particular, we are interested in finding a group whose image by the stable commutator length is Q≥0.

First, we will define its basic properties, then we will furnish some examples and explore its connection with bounded cohomology, via Bavard’s duality theorem, and eventually we will show that in the universal central extension of the Thompson group every rational number is realized as stable commutator length of some element.

2.1 Stable commutator length: definition and first

proper-ties

Throughout the chapter, given a group G we will denote with G0its commutator subgroup, which is the subgroup generated by the elements of the form xyx−1y−1.

Definition 2.1. Let G be a group and let g ∈ G0 be an element of the commutator subgroup. The commutator length of g, denoted with cl(g), is defined as the least number of commutators whose product is g.

We observe that the function g! cl(gn) is subadditive, i.e. cl(gm+n) ≤ cl(gm)+cl(gn). Now, we pass to the stabilization.

Definition 2.2. Let g be an element of G0. The stable commutator length of g, which is denoted by scl(g), is defined as the following limit

scl(g) = lim

n!∞

cl(gn) n .

The limit introduced in Definition 2.2 exists, as a consequence of the following lemma. Lemma 2.3. Let C be a real positive number, and let an be a sequence of non-negative

numbers such that a_n+m ≤ a_n+ a_m+ C. Then the limit of an

n exists.

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Proof. The sequence an

n is non-negative and bounded on both sides, in fact

0 ≤ an n ≤ an−1+ a1+ C n ≤ . . . ≤ na1+ (n − 1)C n ≤ a1+ C.

Now we prove that the limit is equal to the liminf; namely, we will show that for n big enough it holds that an

n ≤ lim inf an

n + , which implies the conclusion. Let > 0. By

definition of liminf, there exists a natural number N > 3C such that aN N < lim inf an n + 3.

Choose n0 > 3 · max{ar|r ∈ N, r ≤ N − 1}. There exist q, s ∈ N such that n0 = qN + s,

with 0 ≤ s ≤ N − 1. Finally, we get an0 n0 ≤ aqN + as+ C n0 ≤ qaN + as+ qC qN + s ≤ aN N + as n0 + C N ≤ lim inf an n + 3 + 3+ 3 = lim inf an n + .

We have already observed that the function n! cl(gn) is non-negative and subaddi-tive, so this implies the existence of the limit.

The very first properties follow immediately from the definition.

Lemma 2.4 (Monotonicity). Let φ : G ! H be a group homomorphism. Then sclH(φ(g)) ≤ sclG(g) for every g ∈ G. Moreover, if φ is an isomorphism, than the

equality holds.

Proof. It follows immediately observing that the image of G0 is contained in H0, since φ is a homomorphism.

Lemma 2.5. Let G be a group, and g ∈ G0 an element of the commutator subgroup. Then there exists a countable subgroup H < G such that sclH(g) = sclG(g). The same

result holds for the commutator length.

Proof. For each n ∈ N, we exhibit gn as a product of cl(g) elements of the commutator subgroup, and we define Hn as the subgroup generated by the elements appearing in

those commutators. The subgroup H will be the one generated by the union of the H_n, namely H = hS

nHni.

2.2 Geometric interpretation of commutator length

If S is an oriented closed connected surface of genus g, we can obtain S by gluing the edges of a 4g-gon P in pairs. This procedure gives us the "standard" presentation of the fundamental group of S, which is

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2.2. GEOMETRIC INTERPRETATION OF COMMUTATOR LENGTH 27 Let X be a topological space, and let αi and βi be elements of π1(X) such that

[α₁, β1] . . . [αg, βg] = id ∈ π1(X).

This equality induces a map S ! X which for each i = 1, . . . , g maps a_i 7! α_i and bi 7! βi; in fact, this association induces a map ∂P ! X (choosing loops in X which

represent the αi and βi) whose image is nullhomotopic in X, hence it extends to a map

P ! X, which in turn descends to a map S ! X. Moreover, any two extensions differ by a pair of maps P ! X which agree on the boundary; these maps define a map S2 ! X, and so an element of π2(X). In other words, we have proved that identities in

the commutator subgroup of π₁(X) correspond to homotopy classes of maps of closed orientable surfaces into X, up to elements of π₂(X).

If S is an oriented closed connected surface of genus g ≥ 1 with one boundary component, its fundamental group is a free group with generators a₁, b1, . . . , ag, bg; the boundary ∂S

(seen as a free loop) represents the conjugacy class of [a₁, b1] . . . [ag, bg]. These surfaces

can be obtained by gluing pairs of edges of a 4g + 1-gon P , leaving one edge "free". Now, let γ be a conjugacy class of an element in π1(X) represented by a loop lγ ⊂ X. If γ

admits a representative in the commutator subgroup, we can write [α₁, β1] . . . [αg, βg] = γ ∈ π1(X).

This defines a map f : ∂P ! X which maps ai 7! αi, bi 7! βi and the free edge to lγ; by

definition of γ, we have that f (∂P ) is null-homotopic in X. By construction, this map factors through the quotient map ∂P ! X induced by gluing all but one edges, and so we can extend it to a map S ! X which maps ∂S to lγ. Summing up, we have shown

that loops corresponding to (conjugacy classes of) elements in the commutator subgroup of π₁(X) bound maps of oriented surfaces with one boundary component into X. Now, we can easily give a topological interpretation of commutator length. Let G be a group and let g ∈ G0 be an element of the commutator subgroup. Let us denote with X a model of the classifying space of G, and by γ the conjugacy class of a loop representing g. Since π1(X) = G, we can interpret cl(g) as the least genus of an oriented closed connected

surface with one boundary component mapping into X with a map f : S ! X so that f (∂S) represents γ. Similarly, we can obtain the stable commutator length estimating the genus of an oriented closed connected surface with a boundary component mapping into X in such a way that its boundary wraps multiple times around γ. However, there is no good reason to focusing only on connected surfaces with only one boundary component. Let S be a compact oriented surface (not necessarily connected), and let us define −χ−(S) as the sum of max(−χ(·), 0) over the connected components of S. Given a topological space X and a loop γ : S1! X we say that a map f : S ! X is admissible if it fits in a commutative diagram ∂S S S1 X. i ∂f f γ

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Since S is oriented, its boundary inherits an orientation, so it makes sense to define its fundamental class, which we denote as usual with [∂S] ∈ H₁(∂S) (we understand that we are working with integer coefficients); we denote also with [S1] ∈ H₁(S1) the fundamental class of S1. We define n(S) as the integer such that

f∗([∂S]) = n(S) · [S1].

By orienting S in an appropriate way, we can guarantee that n(S) ≥ 0. The number n(S) is called total algebraic degree of the map of oriented compact manifolds f : ∂S! S1. Using this notation, we give an intrinsically geometric definition of stable commutator length.

Proposition 2.6. Let π1(X) = G, and let γ : S1 ! X be a free loop representing the

conjugacy class of an element a ∈ G0. Then scl(a) = inf

S

−χ−_(S)

2n(S) ,

where the infimum is taken over all the admissible maps f : S! X as above.

Proof. We have that cl(an) ≤ g if and only if there is an admissible map f : S ! X, where S has exactly one boundary component, such that n(S) = n and −χ−(S) = 2g − 1. Hence,

cl(an₎

n ≥ infS

−χ−_(S)

2n(S) , and an inequality is proved.

On the other hand, let f : S! X be an admissible map. If S has multiple components, at least one of them, let us say Si, satisfies −χ−(Si)/2n(Si) ≤ −χ−(S)/2n(S), so without

loss of generality we can assume that S is connected. Thanks to the multiplicativity of −χ−(·) and 2n(·) under finite covers, we can replace S with any finite cover without changing the ratio −χ−(·)/2n(·), it can be easily shown that we can also assume that S has p ≥ 2 boundary components.

Thanks to a topological lemma (see [7, Lemma 1.12]) we can find a finite cover ρ : S0 ! S of arbitrary large degree N >> 1 such that also S0 has p boundary components; we observe that the map f0 := f ◦ ρ is admissible, and that n(S0) = N · n(S) and −χ−(S0) = N · −χ−(S). Now, we can attach 1-handles to S0 connecting the different boundary components, and extend ∂f0 over these 1-handles with a constant map to a point of S1_.

Each 1-handle that we add increases genus by 1 and reduces the number of boundary components by 1, so it increases −χ−(S0) by 1. Eventually, we obtain a new surface S00 with exactly one boundary component and a map f00satisfying −χ−(S00) = −χ−(S0)+p−1 and n(S00) = n(S0), so

−χ−(S00) 2n(S00) =

p − 1 − N · χ−(S) 2N · n(S) .

Given S, the number of the boundary components p is fixed, but N may be taken as large as we want. So, since S is arbitrary, we can make the right hand side arbitrary

The spectrum of simplicial volume