By Danny Calegari

This publication is a entire creation to the idea of good commutator size, a major subfield of quantitative topology, with large connections to 2-manifolds, dynamics, geometric staff conception, bounded cohomology, symplectic topology, and plenty of different topics. We use confident equipment each time attainable, and concentrate on primary and specific examples. We provide a self-contained presentation of numerous foundational ends up in the idea, together with Bavard's Duality Theorem, the Spectral hole Theorem, the Rationality Theorem, and the valuable restrict Theorem. The contents could be available to any mathematician attracted to those matters, and are awarded with a minimum variety of must haves, yet on the way to functions in lots of components of arithmetic.

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70. It follows that D(rot) ≤ 1 on any subgroup of T . 4. Free products. Bavard Prop. 2 [8] asserts that if G1 and G2 are two groups, and G = G1 ∗ G2 is their free product, then for all nontrivial elements gi ∈ Gi , there is an equality scl(g1 g2 ) = scl(g1 ) + scl(g2 ) + 1/2. This assertion is not quite true as stated. Nevertheless, it turns out that Bavard’s assertion is true when g1 and g2 have infinite order, and can be suitably modified when one or both of them are torsion. We give the correct statement and proof, and defer a discussion of Bavard’s argument and what can be salvaged from it to the sequel.

70 provides a method of calculating scl in some cases, especially when the dimension of Q(G) is small. Given an element a ∈ [G, G], it is natural to ask whether the supremum of φ(a)/D(φ) is realized by some φ ∈ Q(G). 87. Let a ∈ [G, G]. An element φ ∈ Q(G) is extremal for a if scl(a) = φ(a) 2D(φ) The union of 0 with the set of homogeneous quasimorphisms on G which are extremal for a is denoted Qa (G). The next Proposition shows that extremal quasimorphisms always exist. 88. Let a ∈ [G, G]. Then Qa (G) is a nontrivial convex cone in Q(G) which is closed both in the defect and the weak∗ topology.

The boundary map ∂ : C2l1 → C1l1 has a (bounded) cross-section σ defined by the formula 1 1 σ(g) = (g, g) + (g 2 , g 2 ) + · · · 2 4 Proof. The proof is immediate. From this it follows that B1l1 = C1l1 as abstract vector spaces. 61 shows that b B ≤ b 1 for b ∈ C1l1 . Since we also have b B ≥ 31 b 1 , this shows that the quotient norm and the L1 norm on C1l1 are equivalent (though not necessarily isometric). The dual of C1l1 with its L1 norm is Cb1 with its L∞ norm. Dualizing Z2l1 → C2l1 → C1l1 b 1 = (C l1 /Z 2 )′ , if we give shows that the image δ(Cb1 ) is equal to (C2l1 /Z2l1 )′ .