相对非球性研究中的有效有限群
Efficient Finite Groups Arising in the Study of Relative Asphericity
摘要：
我们研究了一类两个生成元的两个关系子群，记为$J_n(m,k)$，它是在研究相对非球面性时作为满足过渡曲率条件的群出现的。这些群的特殊例子出现在文献中，作为有趣阶的有限群。在这里，我们发现无限家庭的非初等虚拟自由群和有限亚阿贝尔非幂零群，我们确定的顺序。所有的梅森素数都是类中非亚循环群的阶的因子，就像所有的素数都是来自其他素数的无穷族一样。我们对有限群进行了同构分类，并证明了我们的类重叠和推广了一类由C.M.Campbell，H.S.M.Coxeter和E.F.Robertson引入的具有三价Cayley图的群$F^{a,b,c}$.周期性提出的群体的理论通知我们的方法，我们扩展了这一理论的一部分（即与多项式结果的连接），以“bicyclically提出的群体”，自然出现在我们的分析。作为我们主要结果的推论，我们得到了新的有限亚循环广义Fibonacci群的无限族。
英文摘要：
We study a class of two-generator two-relator groups, denoted $J_n(m,k)$, that arise in the study of relative asphericity as groups satisfying a transitional curvature condition. Particular instances of these groups occur in the literature as finite groups of intriguing orders. Here we find infinite families of non-elementary virtually free groups and of finite metabelian non-nilpotent groups, for which we determine the orders. All Mersenne primes arise as factors of the orders of the non-metacyclic groups in the class, as do all primes from other conjecturally infinite families of primes. We classify the finite groups up to isomorphism and show that our class overlaps and extends a class of groups $F^{a,b,c}$ with trivalent Cayley graphs that was introduced by C.M.Campbell, H.S.M.Coxeter, and E.F.Robertson. The theory of cyclically presented groups informs our methods and we extend part of this theory (namely, on connections with polynomial resultants) to "bicyclically presented groups" that arise naturally in our analysis. As a corollary to our main results we obtain new infinite families of finite metacyclic generalized Fibonacci groups.
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