# Deformation theory and finite simple quotients of triangle groups I

• Volume: 016, Issue: 7, page 1349-1375
• ISSN: 1435-9855

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## Abstract

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Let $2\le a\le b\le c\in ℕ$ with $\mu =1/a+1/b+1/c<1$ 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 $\left(a,b,c\right)=\left(2,3,7\right)$ and more recently also for general $\left(a,b,c\right)$.) 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 $T$.

## How to cite

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Larsen, Michael, Lubotzky, Alexander, and Marion, Claude. "Deformation theory and finite simple quotients of triangle groups I." Journal of the European Mathematical Society 016.7 (2014): 1349-1375. <http://eudml.org/doc/277325>.

@article{Larsen2014,
abstract = {Let $2 \le a \le b \le c \in \mathbb \{N\}$ 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 [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 $T$.},
author = {Larsen, Michael, Lubotzky, Alexander, Marion, Claude},
journal = {Journal of the European Mathematical Society},
keywords = {triangle groups; representation varieties; finite simple groups; Fuchsian groups; Hurwitz groups; triangle groups; presentations; Fuchsian groups; finite simple groups; representation varieties; Hurwitz groups},
language = {eng},
number = {7},
pages = {1349-1375},
publisher = {European Mathematical Society Publishing House},
title = {Deformation theory and finite simple quotients of triangle groups I},
url = {http://eudml.org/doc/277325},
volume = {016},
year = {2014},
}

TY - JOUR
AU - Larsen, Michael
AU - Lubotzky, Alexander
AU - Marion, Claude
TI - Deformation theory and finite simple quotients of triangle groups I
JO - Journal of the European Mathematical Society
PY - 2014
PB - European Mathematical Society Publishing House
VL - 016
IS - 7
SP - 1349
EP - 1375
AB - Let $2 \le a \le b \le c \in \mathbb {N}$ 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 [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 $T$.
LA - eng
KW - triangle groups; representation varieties; finite simple groups; Fuchsian groups; Hurwitz groups; triangle groups; presentations; Fuchsian groups; finite simple groups; representation varieties; Hurwitz groups
UR - http://eudml.org/doc/277325
ER -

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