Infinite families of isomorphic nonconjugate finitely generated subgroups

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 ${\operatorname {Aut} _\mathbb {Z}}(L{ \otimes _\mathbb {Z}}\mathbb {R},\;\langle \;,\;\rangle \otimes 1)$.