对称群的随机生成子群与图的随机提升
Randomly Generated Subgroups of the Symmetric Group and Random Lifts of Graphs
摘要：
Amit和Linial证明了具有最小度$δ\ge3$的图的随机提升是渐近几乎必然$δ$ -连通的，并提到了将这种概率估计为提升度的函数的问题。我们涉及到一个随机生成的子群的对称群的$n$元素的随机$n$ -升降机的图，并使用它来提供这样的估计以及相关的结果。我们还改进了他们后来的结果显示随机升降机的边缘扩张的下限。我们的证明依赖于新的想法，从群论，使一些改进成为可能。通过证明连通图的随机提升连通的概率等于由$l$随机元生成的对称群的子群传递的概率，精确地计算了具有第一Betti数$l$的连通图的随机提升连通的概率.我们还计算了由$l$随机生成器生成的对称群的环积的子群是传递的概率。我们证明了同伦不变量在随机覆盖图中的存在性，这减少了它们的一些性质，特别是随机正则多重图，并且特别地使得可以计算随机正则多重图连接的确切概率。我们关于随机提升的所有结果都很容易推广到迭代随机提升。
英文摘要：
Amit and Linial showed that a random lift of a graph with minimum degree $δ\ge3$ is asymptotically almost surely $δ$-connected, and mentioned the problem of estimating this probability as a function of the degree of the lift. We relate a randomly generated subgroup of the symmetric group on $n$ elements to random $n$-lifts of a graph and use it to provide such an estimate along with related results. We also improve their later result showing a lower bound on the edge expansion on random lifts. Our proofs rely on new ideas from group theory which make several improvements possible. We exactly calculate the probability that a random lift of a connected graph with first Betti number $l$ is connected by showing that it is equal to the probability that a subgroup of the symmetric group generated by $l$ random elements is transitive. We also calculate the probability that a subgroup of a wreath product of symmetric groups generated by $l$ random generators is transitive. We show the existence of homotopy invariants in random covering graphs which reduces some of their properties to those of random regular multigraphs, and in particular makes it possible to compute the exact probability with which random regular multigraphs are connected. All our results about random lifts easily extend to iterated random lifts.
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