TY - JOUR
T1 - Deformation theory and finite simple quotients of triangle groups I
AU - Larsen, Michael
AU - Lubotzky, Alexander
AU - Marion, Claude
PY - 2014
Y1 - 2014
N2 - Let 2 ≤ a ≤ b ≤ c ∈ ℕ with μ D 1/a + 1/b + 1/c < 1 and let T = Ta;b;c = 〈x, y, z: xa = yb = zc = 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.
AB - Let 2 ≤ a ≤ b ≤ c ∈ ℕ with μ D 1/a + 1/b + 1/c < 1 and let T = Ta;b;c = 〈x, y, z: xa = yb = zc = 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.
KW - Finite simple groups
KW - Representation varieties
KW - Triangle groups
UR - http://www.scopus.com/inward/record.url?scp=84906830964&partnerID=8YFLogxK
U2 - 10.4171/JEMS/463
DO - 10.4171/JEMS/463
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AN - SCOPUS:84906830964
SN - 1435-9855
VL - 16
SP - 1349
EP - 1375
JO - Journal of the European Mathematical Society
JF - Journal of the European Mathematical Society
IS - 7
ER -