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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Infinite families of isomorphic nonconjugate finitely generated subgroups
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by F. E. A. Johnson PDF
Trans. Amer. Math. Soc. 342 (1994), 397-406 Request permission

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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Additional Information
  • © Copyright 1994 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 342 (1994), 397-406
  • MSC: Primary 20E07
  • DOI: https://doi.org/10.1090/S0002-9947-1994-1154542-6
  • MathSciNet review: 1154542