[Paper Review] On deformations of free groups in compact Lie groups

Tsachik Gelander|arXiv (Cornell University)|Nov 27, 2006

Geometric and Algebraic Topology5 references3 citations

TL;DR

This paper proves two long-standing conjectures in geometric group theory and Lie group dynamics: Margulis and Soifer's conjecture on the density of non-virtually free deformations of free groups in compact Lie groups, and Goldman's conjecture on the ergodicity of the Out(Fₙ)-action on Hom(Fₙ, G)/G for n ≥ 3. Using Nielsen transformations and properties of the adjoint representation, the author shows that any n-tuple in a compact Lie group can be perturbed to generate any dense (n−1)-generated subgroup, establishing genericity of non-virtually free images and ergodicity of the outer automorphism group action.

ABSTRACT

We study some properties of the varieties of deformations of free groups in compact Lie groups. In particular we prove a conjecture of Margulis and Soifer about the density of non-virtually free points in such variety, and a conjecture of Goldman on the ergodicity of the action of Aut(Fn) on such variety when n>2.

Motivation & Objective

• To resolve a conjecture by Margulis and Soifer asserting that non-virtually free deformations are dense in the variety of n-tuples generating free groups in compact Lie groups.
• To prove Goldman’s conjecture on the ergodicity of the Out(Fₙ)-action on the space of homomorphisms from Fₙ to a compact Lie group G, modulo inner automorphisms, for n ≥ 3.
• To establish that any n-tuple in a connected compact Lie group can be arbitrarily slightly deformed to generate any given dense (n−1)-generated subgroup, under mild conditions on the subgroup’s minimal generating set.
• To extend these results to the case n = 2 by showing that any pair in a compact non-abelian Lie group can be deformed to generate a group with Serre’s property (FA), hence not virtually free.

Proposed method

• Utilizes Nielsen transformations to perturb n-tuples in Gⁿ into any open subset of Gⁿ, enabling control over the group generated by the deformed tuple.
• Applies the Zariski topology and Burnside’s lemma to show that sets of elements generating the full Lie algebra via the adjoint representation are Zariski open and dense.
• Employs the product map on conjugacy classes C(a) × C(b) → G and proves its differential is surjective under the condition that the centralizers of a and b intersect trivially, using the Killing form and adjoint representation properties.
• Uses the implicit function theorem to deduce that the product map is open near (a,b) when a and b are regular and in general position, ensuring local surjectivity of the product map on conjugacy classes.
• Applies the fact that torsion elements are dense in compact Lie groups to construct deformations within conjugacy classes that preserve density while ensuring the product is torsion.
• Leverages the structure of semisimple Lie groups as almost direct products of simple factors and analyzes projections to each factor to deduce density of the generated subgroup.

Experimental results

Research questions

• RQ1 Can any n-tuple in a connected compact non-abelian Lie group be deformed to generate a non-virtually free group, as conjectured by Margulis and Soifer?
• RQ2 Is the action of Out(Fₙ) on the space Hom(Fₙ, G)/G ergodic for n ≥ 3, as conjectured by Goldman?
• RQ3 For n ≥ 3, is the set of homomorphisms from Fₙ to G whose image is a fixed dense (n−1)-generated subgroup dense in the full space Hom(Fₙ, G)?
• RQ4 Can any pair of elements in a compact Lie group be deformed to generate a group with Serre’s property (FA), implying it is not virtually free?
• RQ5 Under what conditions on the adjoint representation do sets of elements generating the full Lie algebra via Ad(g) become Zariski open and dense?

Key findings

• The set of homomorphisms f ∈ Hom(Fₙ, G) such that f(Fₙ) = Γ is dense in Hom(Fₙ, G) for any dense (n−1)-generated subgroup Γ ≤ G and n ≥ 3.
• For n ≥ 3, any n-tuple in a connected compact Lie group can be arbitrarily slightly deformed to generate any fixed dense (n−1)-generated subgroup Γ, provided Γ is (n−1)-generated.
• The action of Out(Fₙ) on Hom(Fₙ, G)/G is ergodic for n ≥ 3, confirming Goldman’s conjecture.
• For n = 2, any pair in a compact non-abelian Lie group can be deformed to generate a group with Serre’s property (FA), which implies it is not virtually free.
• The product map C(a) × C(b) → G is open near (a,b) when a and b are regular elements with trivially intersecting centralizers, ensuring local surjectivity of the product on conjugacy classes.
• The set of n-tuples generating a dense subgroup in G is open and dense in Gⁿ for n ≥ 3, and the same holds for pairs when G is simple.