# Deformation theory and finite simple quotients of triangle groups I

Michael Larsen; Alexander Lubotzky; Claude Marion

Journal of the European Mathematical Society (2014)

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

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topLarsen, 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 -