# 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

## Abstract

Let $2 \leq a \leq b \leq c \in \mathbb{N}$ with $\mu=1/a+1/b+1/c<1$ and let $T=T_{a,b,c}=\langle x,y,z: x^a=y^b=z^c=xyz=1\rangle$ 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 $T$? (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 \cite{Marionconj} 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 $T$.

## 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