Geodesic
Deformation theory and finite simple quotients of triangle groups I
Journal of the European Mathematical Society, Tome 16 (2014) no. 7, pp. 1349-1375

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Let 2≤a≤b≤c∈N with μ=1/a+1/b+1/c1 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.
DOI : 10.4171/jems/463
Classification : 20-XX, 00-XX
Keywords: Triangle groups, representation varieties, finite simple groups 
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     author = {Michael Larsen and Alexander Lubotzky and Claude Marion},
     title = {Deformation theory and finite simple quotients of triangle groups {I}},
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Michael Larsen; Alexander Lubotzky; Claude Marion. Deformation theory and finite simple quotients of triangle groups I. Journal of the European Mathematical Society, Tome 16 (2014) no. 7, pp. 1349-1375. doi: 10.4171/jems/463

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