Infinite families of isomorphic nonconjugate finitely generated subgroups

Abstract

Let ⟨ , ⟩ : L × L → Z \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 , ⟨ , ⟩ ) ≅ ( L 1 , ⟨ , ⟩ ) ⊥ ( L 2 , ⟨ , ⟩ ) ⊥ ( L 3 , ⟨ , ⟩ ) (L,\;\langle \;,\;\rangle ) \cong ({L_1},\;\langle \;,\;\rangle ) \bot ({L_2},\;\langle \;,\;\rangle ) \bot ({L_3},\;\langle \;,\;\rangle ) in which ( L 1 , ⟨ , ⟩ ) ({L_1},\;\langle \;,\;\rangle ) has signature (2, 1), ( L 2 , ⟨ , ⟩ ) ({L_2},\;\langle \;,\;\rangle ) has signature (n, 1) with n ≥ 2 n \geq 2 , and ( L 3 , ⟨ , ⟩ ) ({L_3},\;\langle \;,\;\rangle ) is either zero or indefinite with rk Z ( L 3 ) ≥ 3 {\text {rk}}_\mathbb {Z}({L_3}) \geq 3 . We show that the integral automorphism group Aut Z ( L , ⟨ , ⟩ ) {\operatorname {Aut} _\mathbb {Z}}(L,\;\langle \;,\;\rangle ) contains an infinite family of mutually isomorphic finitely generated subgroups ( Γ σ ) σ ∈ Σ {({\Gamma _\sigma })_{\sigma \in \Sigma }} , no two of which are conjugate. In the simplest case, when L 3 = 0 {L_3} = 0 , the groups Γ σ {\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 Aut Z ( L ⊗ Z R , ⟨ , ⟩ ⊗ 1 ) {\operatorname {Aut} _\mathbb {Z}}(L{ \otimes _\mathbb {Z}}\mathbb {R},\;\langle \;,\;\rangle \otimes 1) .