Gaps in scl for Amalgamated Free Products and RAAGs

We develop a new criterion to tell if a group G has the maximal gap of 1/2 in stable commutator length (scl). For amalgamated free products G=A⋆CB\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${G = A \star_C B}$$\end{document} we show that every element g in the commutator subgroup of G which does not conjugate into A or B satisfies scl(g)≥1/2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\rm scl}(g) \geq 1/2}$$\end{document}, provided that C embeds as a left relatively convex subgroup in both A and B. We deduce from this that every non-trivial element g in the commutator subgroup of a right-angled Artin group G satisfies scl(g)≥1/2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\rm scl}(g) \geq 1/2}$$\end{document}. This bound is sharp and is inherited by all fundamental groups of special cube complexes. We prove these statements by constructing explicit extremal homogeneous quasimorphisms ϕ¯:G→R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\bar{\phi} : G \to \mathbb{R}}$$\end{document} satisfying ϕ¯(g)≥1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\bar{\phi}(g) \geq 1}$$\end{document} and D(ϕ¯)≤1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${D(\bar{\phi})\leq 1}$$\end{document}. Such maps were previously unknown, even for non-abelian free groups. For these quasimorphisms ϕ¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\bar{\phi}}$$\end{document} there is an action ρ:G→Homeo+(S1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\rho : G \to \mathrm{Homeo}^+(S^1)}$$\end{document} on the circle such that [δ1ϕ¯]=ρ∗eubR∈Hb2(G,R)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${[\delta^1 \bar{\phi}]=\rho^*{\rm eu}^\mathbb{R}_b \in {\rm H}^2_b(G,\mathbb{R})}$$\end{document}, for eubR\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\rm eu}^\mathbb{R}_b}$$\end{document} the real bounded Euler class.


Introduction
For a group G let G be the commutator subgroup. For an element g ∈ G the commutator length (cl(g)) denotes the minimal number of commutators needed to express g as their product. We define the stable commutator length (scl) via scl(g) = lim n→∞ cl(g n )/n. Stable commutator length is well studied and has geometric meaning: Let X be a topological space, let γ be a loop in X and let [γ] be the conjugacy class in π 1 (X) corresponding to γ. Then both cl ([γ]) and scl([γ]) measure the minimal complexity of an orientable surface needed to bound γ. The theory of these invariants is developed by Calegari in [Cal09b]. A group G is said to have a gap in stable commutator length if there is a constant C > 0 such that either scl(g) = 0 or scl(g) ≥ C for every element g ∈ G . If G is non-abelian, such a constant necessarily satisfies C ≤ 1/2. Similarly we may define gaps in scl for classes of groups. Many classes of "negatively curved" groups have a gap in scl; see Subsection 2.2.
A common way of establishing gaps in scl is by constructing quasimorphisms and using Bavard's Duality Theorem (see [Bav91]): For an element g ∈ G , scl(g) = sup φ∈Q(G)φ where Q(G) is the space of homogeneous quasimorphisms and D(φ) is the defect ofφ; see Subsection 2.1 for the definitions and the precise statement. Though it is known that for every element g ∈ G the supremum in Bavard's Duality Theorem is obtained by so-called extremal quasimorphism these maps are only known explicitly in special cases and are hard to construct; see [Cal09a] and [CW11a].
In the first part of this paper, we will construct a family of extremal quasimorphisms on non-abelian free groups. Let F 2 = a, b be the free group on generators a and b and let w ∈ F 2 be such that it does not conjugate into a or b . Then we will construct a homogeneous quasimorphismφ such thatφ(w) ≥ 1 and D(φ) ≤ 1. This realises the well-known gap of 1/2 in the case of non-abelian free groups. Our approach is as follows: instead of constructing more complicated quasimorphismsφ we first "simplify" the element w.
This simplification is formalised by functions Φ : G → A ⊂ F 2 , called letterquasimorphisms; see Definition 4.1. Here A denotes the set of alternating words in F 2 = a, b with the generators a and b. These are words where each letter alternates between {a, a −1 } and {b, b −1 }. Letter-quasimorphisms are a special case of quasimorphisms between arbitrary groups defined by Hartnick-Schweitzer [HS16]. After this simplification, the extremal quasimorphisms on G are obtained by pulling back most basic quasimorphisms F 2 → R via such letter-quasimorphisms G → A ⊂ F 2 . We further deduce that such quasimorphisms are induced by a circle action ρ : G → Homeo + (S 1 ) by examining the defect and using Theorem 2.5 due to Ghys; see also [Ghy87]. We show: Theorem 4.7. Let G be a group, g ∈ G and suppose that there is a letter-quasimorphism Φ : G → A such that Φ(g) is non-trivial and Φ(g n ) = Φ(g) n for all n ∈ N. Then there is an explicit homogeneous quasimorphismφ : G → R such thatφ(g) ≥ 1 and D(φ) ≤ 1.
If G is countable then there is an action ρ : G → Homeo + (S 1 ) such that [δ 1φ ] = ρ * eu R b ∈ H 2 b (G, R), for eu R b the real bounded Euler class.
By Bavard's Duality Theorem it is immediate that if such an element g additionally lies in G , then scl(g) ≥ 1/2. We state Theorem 4.7 separately as it may also be applied in other cases than the ones presented in this paper; see Remark 6.4. Many groups G have the property that for any element g ∈ G there is a letter-quasimorphism Φ g : G → A such that Φ g (g n ) = Φ g (g) n where Φ g (g) ∈ A is non-trivial. We will see that residually free groups and right-angled Artin groups have this property. Note the similarities of this property with being residually free; see Remark 4.9.
In the second part of this paper we apply Theorem 4.7 to amalgamated free products using left-orders. A subgroup H < G is called left-relatively convex if there is an order on the left cosets G/H which is invariant under left multiplication by G. We will construct letter-quasimorphisms G → A ⊂ F 2 using the sign of these orders. We deduce: 200 N. HEUER GAFA Theorem 6.3. Let A, B, C be groups, κ A : C → A and κ B : C → B injections and suppose both κ A (C) < A and κ B (C) < B are left-relatively convex. If g ∈ A C B does not conjugate into one of the factors then there is a homogeneous quasimorphismφ : A C B → R such thatφ(g) ≥ 1 and D(φ) ≤ 1. If G is countable then there is an action ρ : G → Homeo + (S 1 ) such that [δ 1φ ] = ρ * eu R b ∈ H 2 b (G, R), for eu R b the real bounded Euler class. It is possible to generalise Theorem 6.3 to graphs of groups; see Remark 6.4. Again by Bavard's Duality Theorem we infer that any such g which also lies in the commutator subgroup satisfies scl(g) ≥ 1/2. We apply this to right-angled Artin groups using the work of [ADS15]. This way we prove: Theorem 7.3. Every non-trivial element g ∈ G in the commutator subgroup of a right-angled Artin group G satisfies scl(g) ≥ 1/2. This bound is sharp.
This is an improvement of the bound previously found in [FFT16] and [FST17] who deduce a general bound of 1/24, a bound of 1/20 if the right-angled Artin group is two dimensional and a bound of 1/6γ where γ is the chromatic number of the defining graph. Every subgroup of a right-angled Artin group will inherit this bound. Such groups are now known to be an extremely rich class, following the theory of special cube complexes. See [Wis09,HW08,Ago13,Bri13] and [Bri17]. Stable commutator length may serve as an invariant to distinguish virtually special from special cube complexes.
We collect some properties of the quasimorphisms constructed in this paper.
• The quasimorphisms are induced by circle actions ρ : G → Homeo + (S 1 ) even though we do not construct the explicit action ρ. In particular, for every e = g ∈ F where F is a non-abelian free group and scl(g) = 1/2 there is an extremal quasimorphismφ : F → R induced by a circle action. It is unknown if for an arbitrary element g ∈ F there is an action of F on the circle such that the induced quasimorphism is extremal with respect to g. • There are relatively few quasimorphisms needed to obtain the 1/2 bound in Theorem 7.3. Let G be a right-angled Artin group. Analysis of the constructions show that there is a sequence S N ⊂ Q(G) of nested sets of homogeneous quasimorphisms such that for every non-trivial cyclically reduced element g of length less than N there is someφ ∈ S N such thatφ(g) ≥ 1 and D(φ) ≤ 1. We see that |S N | = O(N ) and the rate-constant only depends linearly on the number of generators of the right-angled Artin group. • We obtain gap results even for elements which are not in the commutator subgroup. This suggests that it may be interesting to use Bavard's Dualtiy Theorem as a generalisation of stable commutator length to an invariant of general group elements g ∈ G. That is to study the supremum ofφ(g)/2 wherē φ ranges over all homogeneous quasimorphisms with D(φ) = 1 which vanish or are bounded on a fixed generating set. In [CW11b] the authors studied this supremum over all homogeneous quasimorphisms induced by circle actions.

Quasimorphisms and Bavard's Duality Theorem
In Subsection 2.1 we give basic properties and definitions of stable commutator length and Bavard's Duality Theorem. In Subsection 2.2 we collect some known results on (spectral) gaps in stable commutator length. In Subsection 2.3 we define generalised quasimorphisms and in Subsection 2.4 the well known Brooks quasimorphisms.

Quasimorphisms and Bavard's Duality Theorem. For what follows
Greek letters (α, β) will denote generic functions, upper-case Roman letters (A, B) will denote generic groups, lower-case Roman letters (a, b) generic group elements and code-font (a, b) will denote letters in a free group. We will stick to this notation unless it is mathematical convention to do otherwise. Let G be a group. For two elements g, h ∈ G the commutator is defined via [g, h] = ghg −1 h −1 and the group generated by all such commutators is called the commutator subgroup of G and is denoted by G . For an element g ∈ G we set the commutator length of g. Note that cl is subadditive and hence the limit exists and is called stable commutator length (scl). See [Cal09b] for a comprehensive reference on scl. Calegari showed that in non-abelian free groups scl can be computed 202 N. HEUER GAFA efficiently in polynomial time and is rational. For a group G, the set of possible values of scl is not fully understood, even for non-abelian free groups. See Subsection 2.2 for a discussion on gaps in scl. We note the following basic property: Proposition 2.1. scl is monotone and characteristic. That is, for any group homomorphism θ : G → H and any g ∈ G we have scl(g) ≥ scl(θ(g)). If θ is an automorphism, then scl(g) = scl(θ(g)).
A quasimorphism is a map φ : G → R such that there is a constant D, such that for all g, h ∈ G, |φ(g) The infimum of all such D is called the defect of φ and denoted by D(φ). Note that quasimorphisms form a vector space under pointwise addition and scalar multiplication. A quasimorphismφ is said to be homogeneous ifφ(g n ) = nφ(g) for all n ∈ Z, g ∈ G. In particular,φ is alternating, Every quasimorphism φ : G → R is boundedly close to a unique homogeneous quasimorphismφ : and we callφ the homogenisation of φ. Homogeneous quasimorphisms on G form a vector space, denoted by Q(G).
See Lemma 2.58 of [Cal09b] for a proof. For what follows we will always decorate homogeneous quasimorphisms with a bar-symbol, even if they are not explicitly induced by a non-homogeneous quasimorphism. We refer the reader to [Fri17] and [Cal09b] for references on quasimorphisms and stable commutator length.
If g 1 and g 2 lie in the same conjugacy class of G thenφ(g 1 ) =φ(g 2 ), hence homogeneous quasimorphisms are class functions. The key ingredient to calculate gaps in stable commutator length is Bavard's Duality Theorem: Theorem 2.3 [Bav91]. Let G be a group and let g ∈ G . Then See [Cal09b] for a proof and a generalisation of this statement. This theorem allows us to estimate stable commutator length using (homogeneous) quasimorphisms. It can be shown that the supremum in Bavard's Duality Theorem is obtained. That is, for every element g ∈ G there is a homogeneous quasimorphismφ with D(φ) = 1 such that scl(g) =φ(g)/2. These quasimorphisms are called extremal and were studied in [Cal09a].

(Spectral) Gaps in scl.
It was shown by [DH91] that every non-trivial element w ∈ F n in the commutator subgroup of a non-abelian free group satisfies that scl(w) ≥ 1/2 and that every non-trivial commutator [w 1 , w 2 ] ∈ F n satisfies scl([w 1 , w 2 ]) = 1/2. Using the monotonicity of scl we may conclude that for an arbitrary group G every commutator [g 1 , g 2 ] ∈ G satisfies scl([g 1 , g 2 ]) ≤ 1/2. On the other hand, some elements g ∈ G satisfy scl(g) = 0 for trivial reasons, for example if they are torsions or a positive power of this element is conjugate to a negative power of this element.
We call the infimum of {scl(g) > 0 | g ∈ G } the gap of scl, often called the spectral gap, and say that a group has a gap in scl if this number is positive. Many classes of "negatively curved" groups have a gap in scl.
• Residually free groups have a gap of exactly 1/2 by Duncan and Howie [DH91].
• Mapping class groups of closed orientable surfaces, possibly with punctures, have a gap depending on the surface; see [BBF16]. • Hyperbolic groups have a gap which depends on the hyperbolicity constant and the number of generators; see [CF10]. • Some classes of groups may not have a uniform gap but the first accumulation point on conjugatcy classes of positive scl may be uniformly bounded away from zero. For example for non-elementary, torsion-free hyperbolic groups and for the fundamental groups of closed hyperbolic manifolds this accumulation point is at least 1/12; see Theorem B of [CF10] and see Theorem 3.11 of [Cal09b]. • Sometimes, one may control scl on certain generic group elements. If G = G 1 G 2 is the free product of two torsion-free groups G 1 and G 2 and g ∈ G does not conjugate into one of the factors, then scl(g) ≥ 1/2; see [Che18] and [IK17]. Similarly, if G = A C B and g ∈ G does not conjugate into one of the factors and such that CgC does not contain a copy of any conjugate of g −1 then scl(g) ≥ 1/12. See Theorem D of [CF10] for the first proof of this gap and [CFL16] for the sharp gap and a generalisation to graphs of groups. • Baumslag-Solitar groups have a sharp uniform gap of 1/12; see [CFL16].
Note that this list is not meant to be comprehensive. By monotinicity, having a gap in scl may serve as an obstruction for group embeddings. If H and G are non-abelian groups with H → G and C > 0 is such that every non-trivial element g ∈ G satisfies scl(g) ≥ C then so does every non-trivial element of H .

Generalised Quasimorphisms.
It is possible to generalise quasimorphisms φ : G → R to maps Φ : G → H for G, H arbitrary groups. Two quite different proposals for such a generalisation come from Fujiwara-Kapovich [FK16] and Hartnick-Schweitzer [HS16]. Whereas the former maps are quite restrictive, the latter type of maps are very rich. The "letter-quasimorphisms" defined in this paper will be quasimorphisms as defined by Hartnick- The quasimorphisms G → R constructed in this paper will be all pullbacks of the most basic quasimorphisms F 2 → R via letter-quasimorphisms G → A ⊂ F 2 ; see Remark 4.8.

Brooks Quasimorphisms.
For what follows F 2 will denote the group on two generators a and b. A word w = x 1 · · · x k ∈ F ({a, b}) = F 2 is called reduced if it has no backtracking. Unless stated otherwise we will always assume that elements in the free group are represented by reduced words. A sub-letter Let w ∈ F 2 , g ∈ F 2 be arbitrary reduced words. Let ν w (g) be the number of (possibly overlapping) subwords of w in the reduced word g. Then the function is a quasimorphism, called Brooks quasimorphism. These maps were introduced by Brooks in [Bro81] to show that the vector space of (homogeneous) quasimorphisms of the free group is infinite dimensional. Observe that for a letter x, the map η x is a homomorphism. Brooks quasimorphisms have been vastly generalised to other cases and groups; see [EF97] and [Heu17a].
Let g, h ∈ F 2 and let (c 1 , c 2 , c 3 ) be reduced words such that g = c −1 1 c 2 , h = c −1 2 c 3 , h −1 g −1 = c −1 3 c 1 are reduced words. Then it is easy to see that the value η w (g)+η w (h)−η w (gh) only depends on the first |w|−1 letters of the words c 1 , c 2 , c 3 , hence the defect is indeed finite. There is an extremal Brooks quasimorphism to the basic commutator [a, b], namely η ab − η ba . This and homomorphisms will be the only Brooks quasimorphisms occurring in this paper.

Bounded Cohomology.
We define (bounded) cohomology of discrete groups and state its basic properties. We refer the reader to [Fri17] for a thorough treatment of the bounded cohomology of discrete groups. Let G be a group, let V be a ZG-module and set C n (G, V ) = {f : G n → V }. For what follows, V = Z or V = R and we think of V as a ZG-module with trivial action. Let · ∞ be the l ∞ -norm on C n (G, R) and set Define the well-known coboundary maps for the inhomogeneous resolution δ n : . . , g n+1 ) + · · · · · · (−1) n+1 f (g 1 , . . . , g n ) and note that δ n restricts to δ n : Let φ : G → R be a quasimorphism. Then δ 1 φ ∈ C 2 b (G, R) is a bounded 2-cocycle and hence induces a class [δ 1 φ] ∈ H 2 b (G, R). These classes are exactly the classes which lie in the kernel of the comparison map c 2 : H 2 b (G, R) → H 2 (G, R) described above.

Bounded Cocycles via Actions on the Circle and Vice Versa.
This subsection states a classical correspondence between bounded cohomology and circle actions developed by Ghys; see [Ghy87]. Also, see [BFH16] for a thorough treatment of this topic. Let Homeo + (S 1 ) be the group of orientation preserving actions on the circle and let be the subgroup of the orientation preserving homeomorphisms of the real line that commutes with the integers. By identifying S 1 ∼ = R/Z we obtain a surjection 206 N. HEUER GAFA π : Homeo + Z (R) → Homeo + (S 1 ). The kernel of π is isomorphic to Z via ι : n → f n with f n : x → x + n and lies in the center of Homeo + Z (R). Hence is a central extension and hence corresponds to a class eu ∈ H 2 (Homeo + (S 1 ), Z) the Euler-class. This class is represented by the cocycle ω : (g, h) → σ(g)σ(h)σ(gh) −1 ∈ Z by identifying Z with ker(π) = im(ι) and where σ is any set-theoretic section σ : [Heu17b] for the correspondence of group extensions and bounded cohomology. The image of eu b under the change of . Surprisingly, a converse statement holds: Theorem 2.5. 1 Let G be a discrete countable group and let [ω] ∈ H 2 b (G, Z) be a class represented by a cocycle ω, such that for all g, h ∈ G, ω(g, h) ∈ {0, 1}. Then there is an action ρ : . This allows us to show that certain quasimorphisms are induced by a circle action ρ : G → Homeo + (S 1 ) without explicitly constructing ρ.

Letter-Thin Triples and the Maps α and β
The set of alternating words A ⊂ F 2 is the set of all words in the letters a and b where the letters alternate between {a, a −1 } and {b, b −1 }. For example, aba −1 b −1 is an alternating word but abba −1 b −1 b −1 is not. We will define maps α, β : A → A and develop their basic properties in Subsection 3.1. We also define a version of these maps onĀ 0 , which are conjugatcy classes of even-length words of A to understand how α, β behave on powers; see Proposition 3.9. In Subsection 3.2 we define certain triples (x 1 , x 2 , x 3 ) where x 1 , x 2 , x 3 ∈ A called letter-thin triples. We think of them as the sides of (thin) triangles; see Figure 2. Note that such triples are not triangles in the usual sense, i.e. the sides x 1 , x 2 , x 3 do not correspond to the geodesics between three points in some metric space like a Cayley graph. Letter-thin triples will be crucial in estimating the defect of the quasimorphisms we construct in this paper. We will see that α and β map letter-thin triples to letter-thin triples in Lemma 3.14, which is the main technical result of this paper. In Subsection 3.3 we see that basic Brooks quasimorphisms and homomorphisms behave well on letter-thin triples. We usually prove the properties we state for α, β just for α and note that all properties may be deduced analogously for β by interchanging a and b; see Proposition 3.4, (2).

The Maps α and β, Definition and Properties.
We will describe two maps α, β : A → A sending alternating words to alternating words.
that is, S + a is the set of alternating words which start and end in a and don't contain the letter a −1 and S − a is the set of alternating words which start and end in a −1 and don't contain the letter a. Note that we assume 0 ∈ N, i.e. a ∈ S + a and a −1 ∈ S − a . Analogously we define the sets and observe that b ∈ S + b and b −1 ∈ S − b . We will decompose arbitrary words w ∈ A as a unique product of elements in Proposition 3.1. Let w ∈ A be an alternating word. Then there are y 0 , . . . , y l and s 1 , . . . , s l such that w = y 0 s 1 y 1 s 2 · · · y l−1 s l y l where y i ∈ {b, b −1 } except that y 0 and/or y l may be empty and This expression is unique.
We will call this way of writing w the a-decomposition of w. Analogously, we may also write w ∈ A as We will call this way of writing w the b-decomposition of w. 208 N. HEUER GAFA Proof of Proposition 3.1. Let w ∈ A be an alternating word. Since a ∈ S + a and a −1 ∈ S − a , we may always find some s i ∈ S + a ∪ S − a and some y i ∈ {b, b −1 } such that w = y 0 s 1 y 1 s 2 · · · y n−1 s n y n with possibly y n and / or y 0 empty. Now let m be the minimal n of all such products representing w i.e.
. Set s = s i y i s i+1 and note that s ∈ S + a (resp. s ∈ S − a ). Then w = y 0 s 1 y 1 s 2 · · · y i−1 s y i+1 · · · y m−1 s m y m which would contradict the minimality of m. Hence all s i alternate between S + a and S − a . By comparing two such expressions we see that such an expression is further unique.
Definition 3.2. Let w ∈ A and let w = y 0 s 1 · · · y l−1 s l y l be the a-decomposition of w.
We will formalise and use this behaviour later; see Proposition 3.4 and Proposition 3.8.

GAFA GAPS IN SCL FOR AMALGAMATED FREE PRODUCTS AND RAAGS 209
The images of α and β are obviously contained in the set of alternating words. Moreover, as the s i in the previous definition all alternate between S + a and S − a , none of the consecutive x i have the same sign in the image of α and no consecutive y i have the same sign in the image of β.
Proposition 3.4. The maps α, β : A → A have the following properties: The analogous statement holds for β.
The analogous statement holds for β.
Proof. To see (1), note that if w = y 0 s 1 y 1 · · · y l−1 s l y l is the a-decomposition of w, then The analogous argument holds for β.
Point (2) is evident from the symmetric way α and β have been defined. To see (3), note that α replaces each of the subwords s i by letters a or a −1 . These have size strictly less than |s i | unless s i is the letter a or a −1 already. This shows |α(w)| ≤ |w| with equality only if α(w) = w and it also shows that α • α = α.
For (4) suppose that the a-decomposition of v 1 x is y 1 0 s 1 1 y 1 1 · · · y 1 l1−1 s 1 l1 and the a-decomposition of xv 2 is s 2 1 y 2 1 · · · y 1 l1−1 s 2 l2 y 2 l2 . Both, s 1 l1 and s 2 1 lie in the same set S + a or S − a depending if x = a or x = a −1 . Without loss of generality assume that x = a. The a-decomposition of v 1 xv 2 may be seen to be y 1 0 s 1 1 y 1 1 · · · y 1 l1−1 sy 2 Comparing terms finishes the proposition.
To study how the maps α, β : A → A behave on powers of elements we need to define a version of them on conjugacy classes. LetĀ 0 be the set conjugacy classes of even length alternating words. Note that then necessarily every two representatives w 1 , w 2 ∈ A of the same conjugacy class inĀ 0 are equal up to cyclic permutation of the letters. This is, there are elements v 1 , v 2 ∈ A such that w 1 = v 1 v 2 and w 2 = v 2 v 1 as reduced words. Hence every representative v ∈ A of an element inĀ 0 is automatically reduced.
To see thatᾱ,β :Ā 0 →Ā 0 are well-defined, suppose that w 1 , w 2 ∈ A are both even alternating words which start in a power of a and both represent the same element [w 1 ] = [w 2 ] ∈Ā 0 . Let x 1 , x 2 ∈ {a, a −1 } be the first letters of w 1 and w 2 . Then there are elements v 1 , v 2 ∈ A such that w 1 = x 1 v 1 x 2 v 2 as a reduced word and w 2 = x 2 v 2 x 1 v 1 . Then, by (3) of Proposition 3.4, . This shows thatᾱ is well defined and analogously thatβ is well defined.
The definition ofᾱ given above is useful for performing calculations. However, there is a more geometric way to think aboutᾱ andβ analogous to the definition of α and β. A common way to depict conjugacy classes in the free group is via labels on a circle: Let w = z 1 · · · z n ∈ F 2 be a cyclically reduced word in the letters z i . Then w labels a circle by cyclically labelling the sides of the circle counterclockwise by z 1 , z 2 , . . . , z n so that z n is next to z 1 on the circle. Two cyclically reduced words w ∈ F 2 then yield the same labelling up to rotation if and only if they define the same conjugacy class.
Let [w] ∈Ā 0 be a conjugacy class of a word w ∈ A of even length that contains both at least one a and one a −1 as a subword. We may similarly define an a-decomposition of such a cyclic labelling. One may show that in this geometric model the mapsᾱ (resp.β) can then be defined just like for α and β by replacing the words in S + a by a and the words in S − a by a − . If Example 3.7. Let w = ab −1 a −1 ba −1 b −1 ab −1 ab ∈ A. Its conjugacy class is depicted in Figure 1. We observe that w starts with a and set However, we could have also done an a-decomposition of the elements on a circle as pictured in Figure 1 (a) with s 1 = ab −1 aba ∈ S + a and s 2 = a −1 ba −1 ∈ S − a and obtained the same result. Similarly, let w = abab −1 ab. It's conjugacy class is represented by a cyclic labelling of a circle in Figure 1  Proof. To see thatᾱ,β decrease length unless they fix classes is the same argument as in the proof of Proposition 3.4. If [w] is a non-trivial class in the commutator subgroup of F 2 then there is a reduced representative w such that w = av 1 a −1 v 2 for some appropriate v 1 , v 2 ∈ A and we see thatᾱ([w]) is non-trivial as it also contains the subletters a and a −1 . If w ∈ A is a representative such thatᾱ fixes [w] then w has to be of the form w = k i=1 ay i a −1 y i for some y i , y i ∈ {b, b −1 }, k ≥ 1 and similarly, ifβ fixes a class then the a representative has to be of the form Proof. If w ∈ A does not contain both a positive and a negative power of a, the statement follows by an easy calculation. Note that this is the case if and only ifᾱ([w]) = [e]. Otherwise w contains at least one sub-letter a and one sub-letter a −1 . This is the case if w lies in the commutator subgroup. Suppose without loss of generality that w = v 1 av 2 a −1 v 3 as a reduced word for some v 1 , v 2 , v 3 ∈ A. By multiple applications of Proposition 3.4, we see that as non-reduced elements in the free group. Then we define d 1 , d 2 and w to be the reduced representative of α(c 1 v 1 a)a −1 , α(av 2 a −1 v 3 c 2 ) and α(av 2 a −1 v 3 v 1 a)a −1 , respectively. Moreover, α(av 2 a −1 v 3 v 1 a) is a reduced alternating word which starts and ends in a and contains the a −1 as a sub-letter. If follows that w , the reduced representative of α(av 2 a −1 v 3 v 1 a)a −1 , starts with a, contains a −1 and ends with a power of b, so w is non-empty. Further observe thatᾱ([av 2 a −1 v 3 v 1 ]) is represented by α(av 2 a −1 v 3 v 1 a)a −1 and hence [w ] =ᾱ(w).

Letter-Thin Triples, α and β.
In order to streamline proofs later and ease notation we define an equivalence relation on triples (x 1 , x 2 , x 3 ). We think of such a triple as the sides of a (thin) triangle. We stress that the x i are not actually the side of triangles in some metric space; see Figure 2. Here, we study a special type of triples, namely letter-thin triples in Definition 3.12.
Imagining (x 1 , x 2 , x 3 ) as labelling the sides of a triangle, two triples are equivalent if they may be obtained from each other by a sequence of rotations (i), flips (ii) or by changing the signs of its labels (iii) & (iv).
Proof. The first part is clear from the definitions. Note that α commutes both with "rotating the side" (i) and taking inverses (ii) as α satisfies that α(w −1 ) = α(w) −1 for w ∈ A.
Let w = y 0 s 1 y 1 · · · y k−1 s k y k be the a-decomposition of w (see Definition 3.2), where y i ∈ {b, b −1 } and s i ∈ S + a ∪ S − a alternates between S + a and S − a . Then and s i lie in the same set S + a or S − a . We see that once more, α(φ b (w)) = φ b (α(w)) and hence also α • φ b (x 1 , x 2 , x 3 ) is equivalent to α(x 1 , x 2 , x 3 ). Analogously, we see the statement for β.
For a visualisation of the following definition we refer the reader to Figure 2.
Definition 3.12. Let x 1 , x 2 , x 3 ∈ A be alternating elements. The triple (x 1 , x 2 , x 3 ) is called letter-thin triple in one of the following cases: where all words are required to be reduced.
[T2] There are (possibly trivial) elements c 1 , c 2 ∈ A such that  Similarly, c 2 is either empty or its first letter is a power of a, else the x i would not be alternating. Note that for any letter-thin triple (x 1 , x 2 , x 3 ) of type [T1a] we may always find elements d 1 , d 2 , d 3 ∈ A with first letter a power of b such that where x i ∈ {a, a −1 } are such that not all of x 1 , x 2 and x 3 are equal i.e. have the same parity. As we consider the triples only up to equivalence one may wonder if we can assume that any triple as in Equation (1)  Example 3.13. (a, a, a −1 ) is not letter-thin and by the previous discussion also the triple ( is letter-thin. To see this, note that Note that by definition, if (x 1 , x 2 , x 3 ) is letter-thin then all x 1 , x 2 , x 3 are alternating words.
See Figure 2 for the explanation of the name letter-thin triple: First consider elements g, h ∈ F 2 = a, b . The triple (g, h, (gh) −1 ) corresponds to sides of a geodesic triangle in the Cayley graph Cay(F 2 , {a, b}) with endpoints e, g, gh. Note further that there are words c 1 , c 2 , c 3 ∈ F 2 such that g = c −1 1 c 2 , h = c −1 2 c 3 , (gh) −1 = c −1 3 c 1 and all these expressions are freely reduced. A letter-thin triple (x 1 , x 2 , x 3 ) is such that each x i is in addition alternating and corresponds almost to the sides of a geodesic triangle in a Cayley graph, apart from one letter r ∈ {a, b} in the "middle" of the triangle. Figure 2  Observe that (x 1 , x 2 , x 3 ) is letter-thin if and only if ψ(x 1 , x 2 , x 3 ) is letter-thin for ψ defined as in Proposition 3.4 (2) i.e. ψ is the automorphism ψ : F 2 → F 2 defined via ψ : a → b and ψ : b → a.
Proof. We will proceed as follows: Let (x 1 , x 2 , x 3 ) be a letter-thin triple. By Proposition 3.11 it is enough to check that α(x 1 , x 2 , x 3 ) is letter-thin for one representative of the equivalence class. Hence it suffices to check that α(x 1 , x 2 , x 3 ) is letter thin for 1 a −1 bac 2 , c −1 2 a −1 , ac 1 ). By symmetry, this will show the analogous statement for β.
Proposition 3.4, (4) allows us to compute α piecewise i.e. after each occurrence of a letter a or a −1 in a reduced word. For any reduced word c ∈ A starting with a power of b or being empty, we will write c + for the reduced word represented by a −1 α(ac), which itself is not reduced since α(ac) starts with an a. Similarly, we will write c − for the reduced word represented by aα(a −1 c). Note that c + and c − are either empty or their first letter is a power of b, as α(a ± c) is alternating. If c is a word which already has a subscript, say c i , then we will write c i,+ and c i,− , respectively. We consider each of the above cases independently. For letter-thin triples (x 1 , x 2 , x 3 ) of type [T1a] we compute α(x 1 , x 2 , x 3 ) and we will state exactly which equivalences (i), (ii), (iii) and (iv) of Definition 3.10 are needed to obtain one of the representatives for we will just state the type of α(x 1 , x 2 , x 3 ) without explicitly giving the equivalence.
which is of type [T1b] if c 3,+ is non-trivial and of type [T2b], else. This can be seen analogously to the previous case.
• One of c 1 , c 2 is empty. Up to equivalence and changing indices we may assume that c 2 is empty. Then either − c 1 = ad 1 in which case − c 1 = ad 1 , c 2 = a −1 d 2 in which case This concludes the proof of Lemma 3.14.

Brooks Quasimorphisms, Homomorphisms and Letter-Thin Triples.
For what follows we want to study how the Brooks quasimorphism η 0 = η ab − η ba defined in Example 2.4 or certain homomorphisms behave on letter-thin triples. This will be done in Propositions 3.15 and 3.16, respectively.
Proof. First note that if w = w 1 w 2 ∈ F 2 as a reduced word and if z 1 is the last letter of w 1 and z 2 is the first letter of w, then η 0 (w) = η 0 (w 1 ) + η 0 (z 1 z 2 ) + η 0 (w 2 ). (2) Let (x 1 , x 2 , x 3 ) be a letter-thin triple. Note that the value is invariant under the equivalences (i) and (ii) of Definition 3.10. Up to these equivalences we see that any letter-thin triple (x 1 , x 2 , x 3 ) is equivalent via (i) and (ii) to the following: 220 N. HEUER GAFA • Type [T1a]: (c −1 1 xyc 2 , c −1 2 y −1 xc 3 , c −1 3 x −1 c 1 ), for x ∈ {a, a −1 } and y ∈ {b, b −1 }. If c i is empty set z i = e. Else let z i be the first letter of c i . Then, by using successively Equation (2) we see that Using that η 0 (c −1 ) = −η 0 (c) for any c ∈ F 2 we see that and hence we see that for any choice Else let z i be the first letter of c i . By successively using Equation (2) we see that and again we observe that , for x ∈ {a, a −1 } and y ∈ {b, b −1 }. This case is analogous to the previous case.
Recall that η x : F 2 → Z denotes the homomorphism which counts the letter x.
Proof. Let η be as in the proposition and suppose that (x 1 , x 2 , x 3 ) is letter-thin. Just like in the proof of the previous proposition we will consider the four different types of letter thin triples up to equivalences (i) and (ii) of Definition 3.10.

Gaps via Letter-Quasimorphisms
The aim of this section is to define letter-quasimorphisms and deduce the criterion for 1/2 gaps in scl. There will be two types of letter-quasimorphisms: (general) letter-quasimorphisms (Definition 4.1) and well-behaved letter-quasimorphisms (Definition 4.3). The former is useful for applications, the latter will be useful for proofs. For each letter-quasimorphism Φ : G → A there will be an associated well-behaved letter-quasimorphismΦ : G → A whereΦ(g) is obtained from Φ(g) by modifying its beginning and its end; see Proposition 4.5.

Letter-Quasimorphisms and Well-Behaved Letter-Quasimorphisms.
As always A denotes the set of alternating words of F 2 in the generators a and b.
Definition 4.1. Let G be a group. We say that Φ : G → A is a letter-quasimorphism if Φ is alternating, i.e. Φ(g −1 ) = Φ(g) −1 for every g ∈ G and if for every g, h ∈ G one of the following holds: (2) there are elements c 1 , c 2 , c 3 ∈ A and letters x 1 , x 2 , x 3 such that either x 3 c 1 as freely reduced alternating words.
The motivating example for letter-quasimorphisms is the following: Example 4.2. Consider the map Φ : F 2 → A defined as follows. Suppose that w ∈ F 2 has reduced representation a n1 b m1 · · · a nk b mk with all n i , m i ∈ Z where all but possibly n 1 and / or m k are non-zero. Then set where sign : Z → {+1, 0, −1} is defined as usual. This may be seen to be a letterquasimorphism and will be vastly generalised to amalgamated free products; see Lemma 6.1. Observe that for any group G and any homomorphism Ω : G → F 2 the map Φ • Ω : G → A is a letter-quasimorphism. Suppose that G is residually free. Then for ever non-trivial element g ∈ G there is a homomorphism Ω g : G → F 2 such that Ω g (g) ∈ F 2 is nontrivial. By applying a suitable automorphism on F 2 to Ω g we may assume that Ω g (g) starts in a power of a and ends in a power of b. Then Φ g := Φ • Ω g is a letter quasimorphism such that Φ g (g) is nontrivial and such that Φ g (g n ) = Φ g (g) n .
Definition 4.3. We will call triples (x 1 , x 2 , x 3 ) degenerate if they are equivalent to a triple (w, w −1 , e) for some w ∈ A.
Let G be a group. A map Ψ : G → A is called well-behaved letter-quasimorphism if Ψ is alternating, i.e. Ψ(g −1 ) = Ψ(g) −1 for every g ∈ G, and for all g, h ∈ G, the triple is either letter-thin (see Definition 3.12) or degenerate.
It is easy to see that every well-behaved letter-quasimorphism is also a letterquasimorphism. The contrary does not hold. The map Φ : F 2 → A described in Example 4.2 is a letter-quasimorphism but not a well-behaved letter-quasimorphism. For example for g = a, h = a we obtain (Φ(g), Φ(h), Φ(h −1 g −1 )) = (a, a, a −1 ), which is neither letter-thin nor degenerate.
However, we may assign to each letter-quasimorphism Φ a well-behaved letter-quasimorphismΦ. This will be done by pre-composing Φ with a map w →w defined as follows.
Setw = e whenever w ∈ {a, e, a −1 }. Else let z s be the first and z e be the last letter of w ∈ A. Definew as the reduced element in F 2 freely equal tow := ζ s (z s )wζ e (z e ) where The key point is thatw starts with a and ends with a −1 , unless w ∈ {a, e, a −1 }.
Proposition 4.5. If Φ : G → A is a letter-quasimorphism thenΦ : G → A is a well-behaved letter-quasimorphism, called the associated well-behaved letterquasimorphism.
Proof. As w →w commutes with taking inverses, if Φ is alternating then so isΦ. In what follows we will use the following easy to check claim.
First suppose that g, h are as in Case (1) of Definition 4.1 i.e. Φ(g)Φ(h)Φ(gh) −1 = e. If one of Φ(g), Φ(h) and Φ(gh) are trivial then the two other elements are inverses. Hence, up to rotation and taking inverses we may assume that for some u ∈ A. Hence (ũ,ũ −1 , e) is degenerate.
Both letter-quasimorphisms and well-behaved letter-quasimorphisms are examples of quasimorphism in the sense of Hartnick-Schweitzer [HS16]; see Subsection 2.3. Let Φ be a letter-quasimorphism and letη : F 2 → R be an ordinary homogeneous quasimorphism with defect D which vanishes on the generators a, b. We wish to calculate the defect ofη • Φ. Fix g, h ∈ G. If Φ(g)Φ(h) = Φ(gh), then Else, up to rotating the factors we see that Then, asη is homogeneous η(d −1 1 xd 2 ) =η(xd 2 d −1 1 ) and hence |η(xd 2 d −1 1 ) −η(d 2 d −1 1 )| ≤ D as we assumed that η vanishes on the generators. Then we may estimate and after homogenisation of φ =η • Φ(g) we estimate that D(φ) ≤ 8D using that homogenisation at most doubles the defect; see Proposition 2.2. Hence if Φ(g) ∈ F 2 is such that Φ(g n ) = w n for some non-trivial w ∈ A which also lies in the commutator subgroup F and η : F 2 → R is homogenous and extremal to Φ(g) with defect 1 then, by Bavard, scl(g) ≥φ (g) 16 ≥η (Φ(g)) 16 = scl(Φ(g)) 8 and in particular scl(g) ≥ 1/16. This is already a good estimate but we see that we can do much better; see Theorem 4.7.
We will see that this notion is much more flexible than homomorphisms. There are groups G such that for every non-trivial element g ∈ G there is a letterquasimorphisms Φ such that Φ(g) is non-trivial. This may be possible even if the group G is not residually free, for example if G is a right-angled Artin group; see Section 7.
If G is countable then there is an action ρ : , for eu R b the real bounded Euler class. In particular, the Φ(g 0 ) ∈ A of the Theorem has to be alternating and of even length, else Φ(g 0 ) n would not be an alternating word. Proof. Let Φ : G → A be the letter-quasimorphism as in the theorem and letΦ : G → A be the associated well behaved letter-quasimorphism described above. AsΦ(g 0 ) is obtained from Φ(g 0 ) by just possibly changing the beginning and the end of the word Φ(g 0 ), it is easy to see that there are words c 1 , c 2 , w ∈ A such thatΦ(g n 0 ) = c −1 1 w n−1 c 2 as a freely reduced word for all n ≥ 1. Consider the sequence γ i of maps γ i : A → A defined via γ 0 = id, γ 2k+1 = (α • β) k • α and γ 2k = (β • α) k and note that γ i is either α • γ i−1 or β • γ i−1 ; see Definition 3.2. Analogously define the sequenceγ i :Ā 0 →Ā 0 of maps viaγ 0 = id, γ 2k+1 = (ᾱ •β) k •ᾱ andγ 2k = (β •ᾱ) k and note that everyγ i is eitherᾱ •γ i−1 or β •γ i−1 ; see Definition 3.6. For every letter-thin triple (x 1 , x 2 , x 3 ) also γ i (x 1 , x 2 , x 3 ) is letter-thin by multiple applications of Lemma 3.14. Furthermore, if (x 1 , x 2 , x 3 ) is a degenerate triple as in Definition 4.3, then also Let w be as above and consider the sequenceγ i (w) ∈Ā 0 of conjugacy classes in A 0 . By Proposition 3.8, ifγ i (w) is a non-trivial equivalence class in the commutator subgroup thenγ i+1 (w) either is non-trivial and has strictly smaller word-length or γ i (w) =γ i+1 (w); see also Remark 3.5.

GAPS IN SCL FOR AMALGAMATED FREE PRODUCTS AND RAAGS 229
An order ≺ on a set X is a subset of X × X where we stress that a pair (x, y) ∈ X × X is in this subset by writing x ≺ y. Furthermore, the following holds: • For all x, y ∈ X either x ≺ y or y ≺ x. We have x ≺ y and y ≺ x if and only if x = y. • For all x, y, z ∈ X such that x ≺ y and y ≺ z we have x ≺ z.
A set X with a left group action has a G-invariant order if for all g ∈ G, x 1 , x 2 ∈ X , x 1 ≺ x 2 implies that g.x 1 ≺ g.x 2 . A group G is said to be left orderable if the set G has a G-invariant order with respect to its left action on itself. A subgroup H < G is said to be left relatively convex in G if the G-set G/H has some G-invariant order. Note that this definition is valid even if G itself is not left-orderable. If G itself is orderable, then this is equivalent to the following: There is an order ≺ on G such that for every h 1 , h 2 ∈ H and g ∈ G with h 1 ≺ g ≺ h 2 we may conclude g ∈ H. In this case we simply say that H is convex in G. As e ∈ H, this means that H is a neighbourhood of e. It is not hard to see that left relatively convex is transitive:

Proposition 5.1. 3 Let K < H < G be groups. Then G/K is G-orderable such that H/K is convex if and only if G/H is G-orderable and H/K is H-orderable.
An easy example of a pair H < G such that H is left relatively convex in G is Z < Z 2 embedded in the second coordinate via the standard lexicographic order. Similarly, every subgroup G < Z × G embedded via the second coordinate, is left relatively convex for an arbitrary group G. Every generator of a non-abelian free group generates a left relatively convex subgroup in the total group; see [DH91]. In fact, [ADS15] show that each maximal cyclic subgroup of a right-angled Artin group is left relatively convex.
We wish to state the main Theorem of [ADS15]. For this let T denote an oriented simplicial tree, with vertices V(T) and edges E(T) and two maps ι, τ : E(T) → V(T) assigning to each oriented edge its initial and terminal vertex respectively. Suppose that G acts on T and denote by G v (resp. G e ) the stabilisers of a vertex v ∈ V(T) (resp. an edge e ∈ E(T)). Note that stabilisers of an edge e naturally embed into G ι(e) and G τ (e) .
Theorem 5.2. 4 Suppose that T is a left G-tree such that, for each T-edge e, G e is left relatively convex in G ι(e) and in G τ (e) . Then, for each v ∈ V(T), G v is left relatively convex in G. Moreover, if there exists some v ∈ V(T) such that G v is left orderable, then G is left orderable.
We deduce the following corollary, see Example 19 of [ADS15] using Bass-Serre Theory.
Corollary 5.3. Let A, B and C be groups and let κ A : C → A and κ B : C → B be injections and let G = A C B be the corresponding amalgamated free product