Deformation theory and finite simple quotients of triangle groups I

  • Michael Larsen

    Indiana University, Bloomington, United States
  • Alexander Lubotzky

    Hebrew University, Jerusalem, Israel
  • Claude Marion

    Hebrew University, Jerusalem, Israel


Let 2abcN2 \leq a \leq b \leq c \in \mathbb{N} with μ=1/a+1/b+1/c<1\mu=1/a+1/b+1/c<1 and let T=Ta,b,c=x,y,z:xa=yb=zc=xyz=1T=T_{a,b,c}=\langle x,y,z: x^a=y^b=z^c=xyz=1\rangle 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 TT? (Classically, for (a,b,c)=(2,3,7)(a,b,c)=(2,3,7) and more recently also for general (a,b,c)(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 \cite{Marionconj} showing that various finite simple groups are not quotients of TT, as well as positive results showing that many finite simple groups are quotients of TT.

Cite this article

Michael Larsen, Alexander Lubotzky, Claude Marion, Deformation theory and finite simple quotients of triangle groups I. J. Eur. Math. Soc. 16 (2014), no. 7, pp. 1349–1375

DOI 10.4171/JEMS/463