Homology in Finite Index Subgroups
Homology in Finite Index Subgroups
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有限索引子群中的同源性
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发表时间：
2009
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通讯作者：
L. Wall
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作者：
L. Wall
本文研究如下问题：如果G是有限呈现群，且G > G1 > G2 >…是有限索引子群的降序，那么H1（Gi;Fp）的秩与索引b| G: Gi|相比增长有多快？在包含群G的模p同调和商p群G/H的结构的条件下，我们的一个初步定理给出了指标p的a幂的正规子群H的模p同调的秩的下界。这个界推广了[Lac09b]中Lackenby的一个类似结果，我们用它来强化了该文中从p = 2到所有素数的一个结果。本文的主要结果是关于G是有限体积双曲3流形的基本群，而降序{Gi}上的子群是G上的同余子群。我们研究了无限群SL（2， RP）的结构，其中RP是数域上整数环的p进补全，证明了SL（2， RP）的一致强子群。我们证明了如果G是有限体积双曲3流形的基本群，那么对于R和p的某些选择，它可以用SL（2， RP）来逼近，更准确地说，G的某些同余子群与SL（2， RP）的同余子群之间存在对应关系。我们用它来证明最一般的定理，定理5.3.1。定理5.3.1的一个简单推论是，对于任意的bb bb 0 0，群G有一个有限的索引子群G ‘与同余子群{Gi}，使得H1（Gi;Fp）的秩增长速度至少与bb b1 G ’： Gi bb 5 6−α一样快。
This thesis looks at the following question: If G is a finitely presented group and G > G1 > G2 > . . . is a descending sequence of finite index subgroups, then how fast does the rank of H1(Gi;Fp) grow compared with the index |G : Gi|? One of our preliminary theorems gives a lower bound on the rank of the first mod-p homology of a normal subgroup H of index a power of p, in terms of the mod-p homology of the containing group G and the structure of the quotient p-group G/H . This bound generalizes a similar result of Lackenby in [Lac09b], and we use it to strengthen a result of that paper from the case p = 2 to all primes. The main result of the thesis concerns the case where G is the fundamental group of a finite-volume hyperbolic 3-manifold, and the subgroups in the descending sequence {Gi} are congruence subgroups in G. We study the structure of the profinite group SL(2, RP), where RP is the P-adic completion of a ring of integers in a number field, identifying subgroups of SL(2, RP) which are uniformly powerful. We show that if G is the fundamental group of a finite-volume hyperbolic 3-manifold then it can be approximated by SL(2, RP) for some choices of R and P. More precisely, there is a correspondence between certain congruence subgroups of G and congruence subgroups of SL(2, RP). We use this to prove our most general theorem, Theorem 5.3.1. A simple corollary of Theorem 5.3.1 is that for any ǫ > 0 the group G has a finite index subgroup G′ with congruence subgroups {Gi} such that the rank of H1(Gi;Fp) grows at least as fast as |G′ : Gi| 5 6 −ǫ.