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I keep hearing that fuchsian groups are interesting for other reasons than the Fuchsian model for hyperbolic Riemann surfaces. What are those reasons? Are the Fuchsian groups with fixed points interesting from a geometric perspective? Where do Fuchsian groups appear besides hyperbolic geometry? I also read somewhere about a relation between fuchsian groups and fractals. Does someone know more about that and/or has a good reference? |
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About the relation to fractals: for Fuchsian groups of the first kind, the limit set has Hausdorff dimension 1, i.e., it is not fractal. However, for all other quasifuchsian groups of the first kind, the limit set hat Hausdorff dimension strictly bigger than 1, by a theorem of Rufus Bowen.
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Check out Indra's Pearls. (Mumford, Series, Wright). |
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Fuchsian groups are important for the theory of dessin d'enfants in arithmetic geometry. Wolfart wrote a survey here: http://www.math.uni-frankfurt.de/~wolfart/Artikel/abc.pdf . One typical application is the following: There is some $c>0$, such that for infinitely many $g$ there are $>g^{c\log g}$ non-isomorphic complex curves $C$ of genus $g$ satisfying $|\mathrm{Aut}(C)|=84(g-1)$. For the proof you connect the number of different curves to the number of normal subgroups of the $(2,3,7)$-triangle group, and then show that Fuchsian groups are virtually surface groups, and have therefore many normal subgroups. |
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Fuchsian groups occur naturally in JSJ-theory. In 3-manifolds they occur as the base Groups of the Seifert pieces and in geometric group theory they occur as (quotients of) enclosing groups (also called QH-subgroups). In the case of hyperbolic groups these QH-subgroup carry in some sense the outer automorphism group of the whole group. |
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