By Danny Calegari

This e-book is a entire creation to the speculation of strong commutator size, an enormous subfield of quantitative topology, with gigantic connections to 2-manifolds, dynamics, geometric workforce concept, bounded cohomology, symplectic topology, and plenty of different topics. We use confident tools at any time when attainable, and concentrate on primary and particular examples. We provide a self-contained presentation of numerous foundational leads to the idea, together with Bavard's Duality Theorem, the Spectral hole Theorem, the Rationality Theorem, and the imperative restrict Theorem. The contents could be available to any mathematician attracted to those matters, and are provided with a minimum variety of necessities, yet with a purpose to purposes in lots of parts of arithmetic.

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Then U1 ∪U2 is contained in u1 u2 , so cw (u)−cw (u1 )−cw (u2 ) ≥ 0. Conversely, let U be a maximal disjoint configuration of copies of w in u1 u2 . Then either U contains one copy of w which intersects the juncture, or else it is disjoint from the juncture and decomposes as U = U1 ∪U2 . Hence cw (u)−cw (u1 )−cw (u2 ) ≤ 1 if s = 1 and cw (u) − cw (u1 ) − cw (u2 ) ≤ 0 otherwise. It follows that D(Hw ) ≤ 3(|w| − 1). One cannot do better than O(|w|) in general, as an example like w = abababababa shows.

E. it is a homomorphism. Otherwise, if w = w1 w2 is a reduced expression, hw (w) = 1 whereas hw (w1 ) = hw (w2 ) = 0. This proves the first statement. Let u, v ∈ F be reduced. Then we can uniquely write u = u′ x, v = x−1 v ′ where ′ ′ u v is the reduced representative of uv. Let s1 , s2 , s3 be the signs of the reduced expressions u′ x, x−1 v ′ , u′ v ′ respectively. We calculate hw (uv) − hw (u) − hw (v) = hw (uv) − hw (u) − hw (v) − hw (u′ ) + hw (u′ ) − hw (v ′ ) + hw (v ′ ) + hw (x) − hw (x−1 ) = (0 or s3 ) − (0 or s1 ) − (0 or s2 ) After possibly replacing w with w−1 and are only nine possibilities for (s1 , s2 , s3 ):  0 for    1 for |hw (uv) − hw (u) − hw (v)| ≤ 2 for    3 for reversing the order of the strings, there (0, 0, 0) (1, 0, 0), (0, 0, 1), (1, −1, 0), (1, 0, 1) (1, 1, 0), (1, 1, 1), (1, 0, −1) (1, 1, −1) Case ((1, 0, −1)).

In other words, SP admits a hyperbolic metric. Moreover, the only disk pieces are components of Si mapping with degree a multiple of ni to γi , in the case gi is torsion. Since fP is proper, it has a well-defined degree. Since fP−1 (γP ) is equal to ∂S, the degree is n.

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