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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Infinite families of isomorphic nonconjugate finitely generated subgroups

Author: F. E. A. Johnson
Journal: Trans. Amer. Math. Soc. 342 (1994), 397-406
MSC: Primary 20E07
MathSciNet review: 1154542
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Abstract: Let $ \langle \;,\;\rangle :L \times L \to \mathbb{Z}$ be a nondegenerate symmetric bilinear form on a finitely generated free abelian group L which splits as an orthogonal direct sum $ (L,\;\langle \;,\;\rangle ) \cong ({L_1},\;\langle \;,\;\rangle ) \bot ({L_2},\;\langle \;,\;\rangle ) \bot ({L_3},\;\langle \;,\;\rangle )$ in which $ ({L_1},\;\langle \;,\;\rangle )$ has signature (2, 1), $ ({L_2},\;\langle \;,\;\rangle )$ has signature (n, 1) with $ n \geq 2$, and $ ({L_3},\;\langle \;,\;\rangle )$ is either zero or indefinite with $ {\text{rk}}_\mathbb{Z}({L_3}) \geq 3$. We show that the integral automorphism group $ {\operatorname{Aut} _\mathbb{Z}}(L,\;\langle \;,\;\rangle )$ contains an infinite family of mutually isomorphic finitely generated subgroups $ {({\Gamma _\sigma })_{\sigma \in \Sigma }}$, no two of which are conjugate. In the simplest case, when $ {L_3} = 0$, the groups $ {\Gamma _\sigma }$ are all normal subdirect products in a product of free groups or surface groups. The result can be seen as a failure of the rigidity property for subgroups of infinite covolume within the corresponding Lie group $ {\operatorname{Aut} _\mathbb{Z}}(L{ \otimes _\mathbb{Z}}\mathbb{R},\;\langle \;,\;\rangle \otimes 1)$.

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