Determining Fuchsian groups by their finite quotients
Determining Fuchsian groups by their finite quotients
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通过有限商确定 Fuchsian 群
DOI：
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发表时间：
2014
期刊：
影响因子：
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通讯作者：
A. Reid
中科院分区：
文献类型：
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作者：
M. Bridson;M. Conder;A. Reid
设C（Γ）是有限群的同构类的集合，这些有限群是Γ的同态象。我们研究了当Γ是一个几何兴趣群时C（Γ）确定Γ的程度。如果Γ 1是PSL（2，R）中的格，Γ 2是任何连通李群中的格，则C（Γ 1）= C（Γ 2）意味着Γ 1 <$Γ 2。若F是自由群，且Γ是直角Artin群或剩余自由群（有一个额外条件），则C（F）= C（Γ）蕴含F <$Γ。若Γ 1 <PSL（2，C）和Γ 2 <G是非一致算术格，其中G是一个具有平凡中心且无紧因子的半单李群，则C（Γ 1）= C（Γ 2）蕴涵G ∈ PSL（2，C）且Γ 2属于任意多个可积类中的一个.这些结果证明了使用profinite群的理论，我们没有表现出明确的有限的等价物，区分群体的问题。但在特殊情况下，两个非同构的三角形群，我们给出了一个明确的描述，有限的同胚，以区分他们。
Let C(Γ) be the set of isomorphism classes of the finite groups that are quotients (homomorphic images) of Γ. We investigate the extent to which C(Γ) determines Γ when Γ is a group of geometric interest. If Γ1 is a lattice in PSL(2, R) and Γ2 is a lattice in any connected Lie group, then C(Γ1) = C(Γ2) implies that Γ1 ≅ Γ2. If F is a free group and Γ is a right-angled Artin group or a residually free group (with one extra condition), then C(F) = C(Γ) implies that F ≅ Γ. If Γ1 < PSL(2, C) and Γ2 < G are nonuniform arithmetic lattices, where G is a semisimple Lie group with trivial centre and no compact factors, then C(Γ1) = C(Γ2) implies that G ≅ PSL(2, C) and that Γ2 belongs to one of finitely many commensurability classes. These results are proved using the theory of profinite groups; we do not exhibit explicit finite quotients that distinguish among the groups in question. But in the special case of two non-isomorphic triangle groups, we give an explicit description of finite quotients that distinguish between them.