论文标题
通用长度在可数组上起作用
Generic length functions on countable groups
论文作者
论文摘要
令$ l（g）$表示在具有点侧收敛拓扑的可数组$ g $上的整数值长度函数的空间。假设$ g $不满足任何非平凡的混合身份，我们证明$ g $上的通用（在baire类别中）长度函数是一个单词长度，而关联的cayley图对$ g $独立的某些通用图$ u $是同构。另一方面，我们证明了$ l（g）$的每个合并子集包含$ 2^{\ aleph_0} $无与伦比的长度函数。这些结果的组合产生$ 2^{\ aleph_0} $成对的非等效表示$ g \ to aut（u）$。我们还证明，从模型理论的角度来看，通用长度函数实际上是无法区分的。 $ g $ $ g $在$ l（g）$上的拓扑传递在后一个结果的证明中起着至关重要的作用。
Let $L(G)$ denote the space of integer-valued length functions on a countable group $G$ endowed with the topology of pointwise convergence. Assuming that $G$ does not satisfy any non-trivial mixed identity, we prove that a generic (in the Baire category sense) length function on $G$ is a word length and the associated Cayley graph is isomorphic to a certain universal graph $U$ independent of $G$. On the other hand, we show that every comeager subset of $L(G)$ contains $2^{\aleph_0}$ asymptotically incomparable length functions. A combination of these results yields $2^{\aleph_0}$ pairwise non-equivalent regular representations $G\to Aut(U)$. We also prove that generic length functions are virtually indistinguishable from the model-theoretic point of view. Topological transitivity of the action of $G$ on $L(G)$ by conjugation plays a crucial role in the proof of the latter result.
