Contracting elements and random walks
Contracting elements and random walks
复制标题
DOI：
10.1515/crelle-2015-0093
复制
发表时间：
2011-12
期刊：
影响因子：
--
通讯作者：
A. Sisto
中科院分区：
文献类型：
--
作者：
A. Sisto
我们定义了群的压缩元的新概念，证明了压缩元与相对双曲群中的双曲元、映射类群中的伪Anosov、在真{mathm{cat}(0)}空间上适当作用的群中的秩1等距、图流形群中的Bass-Serre树上的元一致.我们还定义了一个相关的弱压缩元的概念，并证明了它们与作用在双曲空间上的群中的双曲元以及与{\mathm{out}(F_(n}))}，{n\geq3}中的iwip重合。我们证明了每个弱压缩元都包含在一个双曲嵌入的基本子群中，这使得我们可以回答[16]中的一个问题。我们证明了包含(弱)压缩元的非初等有限生成子群中的任何简单随机游动都以指数衰减的概率结束于一个非(弱)压缩元。
We define a new notion of contracting element of a group and we show that contracting elements coincide with hyperbolic elements in relatively hyperbolic groups, pseudo-Anosovs in mapping class groups, rank one isometries in groups acting properly on proper {\mathrm{CAT}(0)} spaces, elements acting hyperbolically on the Bass–Serre tree in graph manifold groups. We also define a related notion of weakly contracting element, and show that those coincide with hyperbolic elements in groups acting acylindrically on hyperbolic spaces and with iwips in {\mathrm{Out}(F_{n})} , {n\geq 3} . We show that each weakly contracting element is contained in a hyperbolically embedded elementary subgroup, which allows us to answer a problem in [16]. We prove that any simple random walk in a non-elementary finitely generated subgroup containing a (weakly) contracting element ends up in a non-(weakly-)contracting element with exponentially decaying probability.