群与代数中的共长性与本质性
Cogrowth and essentiality in groups and algebras
摘要：
子群的共增长定义为一组具有最小长度的陪集代表的增长。一个子群是本质的，如果它与每个非平凡的平凡子群非平凡相交。本文的主要结果是：每一个严格递增但至多为指数函数的函数$f:{\Bbb N}\cup \{0\}\rightarrow {\Bbb N}$等价于秩为2的自由群的无穷指数本质子群的共生函数.这类函数适当地包含群的增长函数类。引入了代数中右理想的增长性和共增长性的概念。我们证明了当代数没有零因子时，每一个共增长小于代数的共增长的右理想都是本质的。
英文摘要：
The cogrowth of a subgroup is defined as the growth of a set of coset representatives which are of minimal length. A subgroup is essential if it intersects non-trivially every non-trivial subgroup. The main result of this paper is that every function $f:{\Bbb N}\cup \{0\}\rightarrow {\Bbb N}$ which is strictly increasing, but at most exponential, is equivalent to a cogrowth function of an essential subgroup of infinite index of the free group of rank two. This class of functions properly contains the class of growth functions of groups. The notions of growth and cogrowth of right ideals in algebras are introduced. We show that when the algebra is without zero divisors then every right ideal, whose cogrowth is less than that of the algebra, is essential.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
