Deformation theory and finite simple quotients of triangle groups I

2013/01/14 by Michael Larsen, Alexander Lubotzky, Larsen, Michael +3
Mathematics · #FOS: Mathematics #Finite Group Theory Research #Geometric and Algebraic Topology #Group Theory (math.GR) #Homotopy and Cohomology in Algebraic Topology #math.GR

paper · pdf · doi:10.48550/arxiv.1301.2949

arxiv created 2013/01/14 · openalex publication_date 2013/01/14 · arxiv updated 2013/01/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let 2 ≤ a ≤ b ≤ c ∈ ℕ with μ=1/a+1/b+1/c<1 and let T=T_a,b,c=< x,y,z: x^a=y^b=z^c=xyz=1> be the corresponding hyperbolic triangle group. Many papers have been dedicated to the following question: what are the finite (simple) groups which appear as quotients of T? (Classically, for (a,b,c)=(2,3,7) and more recently also for general (a,b,c).) These papers have used either explicit constructive methods or probabilistic ones. The goal of this paper is to present a new approach based on the theory of representation varieties (via deformation theory). As a corollary we essentially prove a conjecture of Marion [21] showing that various finite simple groups are not quotients of T, as well as positive results showing that many finite simple groups are quotients of T.

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