Stable commutator length in word-hyperbolic groups
Stable commutator length in word-hyperbolic groups
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DOI：
10.4171/ggd/75
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发表时间：
2006-11
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通讯作者：
Danny Calegari;K. Fujiwara
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文献类型：
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作者：
Danny Calegari;K. Fujiwara
在本文中，我们获得了字双曲群和作用于双曲空间的某些群（即作用于复曲线的映射类群，以及作用于关联的 Bass-Serre 树的合并自由积）中元素的稳定交换器长度的统一正下界。如果 G 是字双曲群，且相对于对称生成集 S 是 δ 双曲，则存在仅取决于 δ 和 |S| 的正常数 C使得 G 的每个元素要么具有与其倒数共轭的幂，要么该元素的稳定换向器长度至少等于 C。根据巴瓦尔定理，稳定换向器长度的这些下界意味着存在对缺陷进行统一控制的拟同构；然而，我们展示了如何直接构造这样的拟同构。我们还证明了此类群中元素族的各种分离定理，构造齐次拟同构（同样具有统一估计），其在某些指定元素上为正，而在平移长度统一有界的某些独立元素族上消失。最后，我们证明了无扭字双曲群中稳定换向器长度的第一个累加点包含在1/12和1/2之间。这给出了双曲群中的共轭类具有小的稳定交换子长度意味着什么的普遍意义，并且可以被认为是一种“同调马古利斯引理”。
In this paper we obtain uniform positive lower bounds on the stable commutator length of elements in word-hyperbolic groups and certain groups acting on hyperbolic spaces (namely the mapping class group acting on the complex of curves, and an amalgamated free product acting on an associated Bass-Serre tree). If G is a word-hyperbolic group that is δ-hyperbolic with respect to a symmetric generating set S, then there is a positive constant C depending only on δ and on |S| such that every element of G either has a power which is conjugate to its inverse, or else the stable commutator length of the element is at least equal to C. By Bavard’s theorem, these lower bounds on stable commutator length imply the existence of quasimorphisms with uniform control on the defects; however, we show how to construct such quasimorphisms directly. We also prove various separation theorems on families of elements in such groups, constructing homogeneous quasimorphisms (again with uniform estimates) which are positive on some prescribed element while vanishing on some family of independent elements whose translation lengths are uniformly bounded. Finally, we prove that the first accumulation point for stable commutator length in a torsion-free word-hyperbolic group is contained between 1/12 and 1/2. This gives a universal sense of what it means for a conjugacy class in a hyperbolic group to have a small stable commutator length, and can be thought of as a kind of “homological Margulis lemma”.