Download scl by Danny Calegari PDF

By Danny Calegari

This ebook is a accomplished advent to the idea of sturdy commutator size, a major subfield of quantitative topology, with mammoth connections to 2-manifolds, dynamics, geometric crew thought, bounded cohomology, symplectic topology, and plenty of different topics. We use confident equipment at any time when attainable, and concentrate on primary and particular examples. We provide a self-contained presentation of numerous foundational ends up in the speculation, together with Bavard's Duality Theorem, the Spectral hole Theorem, the Rationality Theorem, and the relevant restrict Theorem. The contents might be available to any mathematician drawn to those matters, and are offered with a minimum variety of must haves, yet that allows you to purposes in lots of parts of arithmetic.

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On the other hand, for fixed m, there is an equality ani ) ani ) + (m − 1) n→∞ n→∞ n n the right hand limit exists by the subadditivity of cln,m , and therefore the left hand side does too. 6. STABLE COMMUTATOR LENGTH AS A NORM 37 Given a space X and loops γi : S 1 → X we say that a map f : S → X is admissible if there is a commutative diagram: i . ∂S ................................................................................. S ... . ... ∂f............. f............. . i S .... 1 .............................................................................

For example, if a = [x, y] then ∂((xyx−1 , x) + ([x, y], y) − (x, y)) = [x, y] In general, a one-vertex triangulation of a surface of genus g with one boundary component exhibits a product of g commutators as an element of B1 . 53. Let a ∈ B1 (G; R). The Gersten boundary norm (or just the Gersten norm or the boundary norm) of a, denoted a B , is defined by a B = inf ∂A=a A 1 where the infimum is taken over all 2-chains A ∈ C2 (G; R) with boundary a, and A 1 denotes the usual L1 norm. 54. Gersten calls his norm a filling norm in [90].

8]). There is an equality 1 scl(a) = fill(a) 4 Proof. An expression of an as a product of commutators lets us construct an orientable surface S with one boundary component, and a homomorphism ϕ : π1 (S) → G with ϕ∗ ∂S = an in π1 . 5. BAVARD’S DUALITY THEOREM 35 Conversely, let A be a chain with ∂A = a with A 1 close to a B . Let V be the finite dimensional subspace of C2 (G; R) consisting of 2-chains with support contained in the support of A. 5). So we may assume A is rational, after changing its norm an arbitrarily small amount.

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