On sequences of finitely generated discrete groups
On sequences of finitely generated discrete groups
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DOI：
10.1090/conm/510/10024
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发表时间：
2007-08
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通讯作者：
M. Kapovich
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作者：
M. Kapovich
考虑秩为1的李群G的离散子群序列Gamma_i=rho_i（Gamma），其中rho_i的表示不一定是忠实的.证明了对于代数收敛序列（Gamma_i），除非Gamma_i（最终）是初等的或包含任意高阶的正规有限子群，否则其代数极限是G的一个离散非初等子群.在发散序列（Gamma_i）的情形下，证明了真实的树T上的极限作用满足一定的半稳定性条件，推广了Rips引入的稳定性概念.然后，我们验证组伽玛分裂为一个汞合金或HNN扩展生成的组，使边缘组有一个顺从的图像在等距组的T。
We consider sequences of finitely generated discrete subgroups Gamma_i=rho_i(Gamma) of a rank 1 Lie group G, where the representations rho_i are not necessarily faithful. We show that, for algebraically convergent sequences (Gamma_i), unless Gamma_i's are (eventually) elementary or contain normal finite subgroups of arbitrarily high order, their algebraic limit is a discrete nonelementary subgroup of G. In the case of divergent sequences (Gamma_i) we show that the limiting action on a real tree T satisfies certain semistability condition, which generalizes the notion of stability introduced by Rips. We then verify that the group Gamma splits as an amalgam or HNN extension of finitely generated groups, so that the edge group has an amenable image in the isometry group of T.