半群上随机游动的共生性、顺从性和谱半径
On cogrowth, amenability and the spectral radius of a random walk on a semigroup
摘要：
我们引入了两个自然的概念的共同增长的生成半群-一个本地和一个全球-和研究它们的关系，顺从性和随机游动。我们建立了最小和最大可能的值的共生率，并表明，非单演自由半群的确切特点是最小的全球共生。我们考虑了群的共生性和半群的顺从性的关系。我们还研究了随机游动的关系，特别是$\ell_2$ -空间上的相关马尔可夫算子（定义时）的谱半径。我们证明了无论是最大整体共生还是弱Følner条件都足以使它的谱半径至少为$1$，因为左顺从性蕴涵着弱Følner条件，这代表了群的Kesten定理的一个蕴涵在半群上的推广.结合已知的顺从性结果，我们能够在广泛的半群类中建立一些新的（左或右）顺从性的充分条件。特别地，极大局部左共增长在任何左可逆半群中蕴含顺从性，而极大全局共增长（这是一个弱得多的性质）足以在一个非常广泛的半群类中包含所有逆半群、左可逆左可消半群和左可逆正则半群。
英文摘要：
We introduce two natural notions of cogrowth for finitely generated semigroups --- one local and one global --- and study their relationship with amenability and random walks. We establish the minimal and maximal possible values for cogrowth rates, and show that non-monogenic free semigroups are exactly characterised by minimal global cogrowth. We consider the relationship with cogrowth for groups and with amenability of semigroups. We also study the relationship with random walks on finitely generated semigroups, and in particular the spectral radius of the associated Markov operators (when defined) on $\ell_2$-spaces. We show that either of maximal global cogrowth or the weak Følner condition suffices for its spectral radius to be at least $1$, since left amenability implies the weak Følner condition, this represents a generalisation to semigroups of one implication of Kesten's Theorem for groups. By combining with known results about amenability, we are able to establish a number of new sufficient conditions for (left or right) amenability in broad classes of semigroups. In particular, maximal local cogrowth left implies amenability in any left reversible semigroup, while maximal global cogrowth (which is a much weaker property) suffices for left amenability in an extremely broad class of semigroups encompassing all inverse semigroups, left reversible left cancellative semigroups and left reversible regular semigroups.
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