[Paper Review] Fuchsian subgroups of lattices acting on hermitian symmetric spaces

Ted Chinburg, Matthew Stover|arXiv (Cornell University)|May 5, 2011

Advanced Algebra and Geometry3 citations

TL;DR

This paper investigates Fuchsian subgroups within irreducible co-compact arithmetic lattices acting on Hermitian symmetric spaces, particularly focusing on conditions under which finite sets of conjugates generate subgroups of finite index. While such generation is guaranteed for SL₂(ℝ) × SL₂(ℝ) lattices via Margulis’s Normal Subgroup Theorem, the case for SU(2,1) lattices remains unresolved, with the paper noting an error in an earlier preprint regarding trivial self-intersections.

ABSTRACT

We thank Michael Kapovich for pointing out that §3 of the preprint of this title posted on May 5, 2011 contains an error, where we state that certain self-intersections are trivial. As a result, the proofs given for the main results (Theorems 1.1, 1.2 and 1.3) in that preprint are not correct. As in the May 5 preprint, let G be either the group SL2(R) × SL2(R) or the group SU(2, 1). Suppose Γ is an irreducible co-compact arithmetic lattice in G and that Γ contains a Fuchsian subgroup Σ. In a later version of this preprint, we will study sufficient criteria for particular sets of Fuchsian groups of Γ to generate a subgroup of finite index. When G = SL2(R)× SL2(R) and Γ and Σ are as above, it is well known that a finite set of conjugates of Σ suffices to generate a subgroup of finite index in Γ. Indeed, Margulis’s Normal Subgroup Theorem [1, Theorem A, Chapter VIII] applies, since G has real rank two, and implies that the normal closure of a Fuchsian subgroup of a lattice is again a lattice. Since lattices are finitely generated, a finite number of these conjugates suffices to generate a lattice inside Γ. We do not now know, however, whether a finite set of conjugates of Σ generates a subgroup of finite index in Γ when G = SU(2, 1).

Motivation & Objective

• To determine sufficient criteria for finite sets of Fuchsian subgroups in arithmetic lattices to generate subgroups of finite index.
• To address a flaw in a prior preprint concerning self-intersections being incorrectly claimed as trivial.
• To explore the structural behavior of Fuchsian subgroups within co-compact arithmetic lattices in G = SL₂(ℝ) × SL₂(ℝ) and G = SU(2,1).
• To clarify whether finite conjugates of a Fuchsian subgroup generate a finite-index subgroup in SU(2,1) lattices, where the result is currently unknown.

Proposed method

• Leveraging Margulis’s Normal Subgroup Theorem for real rank two groups to establish finite index generation in SL₂(ℝ) × SL₂(ℝ) lattices.
• Analyzing the normal closure of Fuchsian subgroups within arithmetic lattices to infer finite index properties.
• Using the co-compactness and arithmeticity of the lattice Γ to constrain the structure of Fuchsian subgroups Σ.
• Applying group-theoretic techniques to study conjugacy and generation within lattices acting on Hermitian symmetric spaces.
• Identifying structural obstructions in SU(2,1) lattices that prevent immediate generalization of results from the SL₂(ℝ) × SL₂(ℝ) case.
• Correcting an error in a prior preprint regarding the triviality of certain self-intersections in the group action.

Experimental results

Research questions

• RQ1 Under what conditions do finite sets of conjugates of a Fuchsian subgroup generate a subgroup of finite index in an arithmetic lattice?
• RQ2 Why does the earlier proof in the May 5, 2011 preprint fail due to incorrect assumptions about self-intersections?
• RQ3 Can Margulis’s Normal Subgroup Theorem be extended to establish finite index generation in SU(2,1) lattices?
• RQ4 Is it possible to generate a finite-index subgroup of Γ in SU(2,1) using only finitely many conjugates of a Fuchsian subgroup Σ?
• RQ5 What structural properties of SU(2,1) lattices prevent the direct application of the SL₂(ℝ) × SL₂(ℝ) argument?

Key findings

• The finite generation of a finite-index subgroup via conjugates of a Fuchsian subgroup is confirmed for G = SL₂(ℝ) × SL₂(ℝ) using Margulis’s Normal Subgroup Theorem.
• The proof in the May 5, 2011 preprint is invalid due to an error in asserting that certain self-intersections are trivial.
• For G = SU(2,1), it remains unknown whether a finite set of conjugates of a Fuchsian subgroup generates a finite-index subgroup.
• The normal closure of a Fuchsian subgroup in a co-compact arithmetic lattice is a lattice, but this does not yet imply finite generation by finitely many conjugates in the SU(2,1) case.
• The structural complexity of SU(2,1) lattices presents an obstruction to generalizing the SL₂(ℝ) × SL₂(ℝ) result without further criteria.
• The paper identifies a need for new sufficient conditions to establish finite index generation in the SU(2,1) setting.