By Henry Berns, Henry Berro
Книга содержит конкретные инструкции и шаблоны для почти 50 трехмерных проектовминиатюрных трехмерных моделей животных, людей и прочих. Фигурки выпиливаются из прямоугольных брусков с двух сторон и обтачиваются до окончательных форм. Книга для любителей выпиливания.
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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. 42 2. STABLE COMMUTATOR LENGTH Proof. 60 that there is an isomorphism of vector spaces Q/H 1 ∼ = (Z2l1 /Z 2 )′ . As a dual space, we can endow Q/H 1 with the weak∗ topology.
2  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.
Let S be a surface with one boundary component, and f : S → X a map sending ∂S to γP with degree n. We have scl(g1 g2 ) ≤ −χ(S)/2n. Make f transverse to the γi . The surface is decomposed into pieces, which are the closures, in the path topology, of S − f −1 (γ1 ∪ γ2 ). We say that f is efficient if no piece has a boundary component which maps with degree zero to a γi , and if no piece is an annulus with both boundary components mapping to the same γi with opposite degree. If S is not efficient, the Euler characteristic of S can be increased by surgering S along a circle which maps to some γi with degree 0 (and is therefore null-homotopic), or simplified by homotoping a trivial annulus.