# Why are Fuchsian groups interesting?

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?

• The tesselation of the upper-half plane is fractal in nature – reuns Jan 21 '17 at 22:33
• The tesselation is certainly not fractal. What might be fractal are the Limit Sets, although not in your example where the Limit set is just a Circle. – ThiKu Jan 21 '17 at 23:22

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 has Hausdorff dimension strictly bigger than 1, by a theorem of Rufus Bowen. Check out Indra's Pearls. (Mumford, Series, Wright).

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.

• Relatedly, if the Gromov boundary of a hyperbolic group $G$ is a circle, then the Convergence Group Theorem of Casson--Jungreis--Gabai--Tukia asserts that $G$ is finite-by-Fuchsian. – HJRW Jan 23 '17 at 7:07

Fuchsian groups, particularly those which are cocompact, form the tips of several big mathematical icebergs. To put this less metaphorically, several discoveries about Fuchsian groups, obtained using hyperbolic geometry, led to generalizations which fueled the growth of combinatorial and geometric group theory in the 20th century.

By work of Dehn and others, cocompact Fuchsian groups are early examples of geometric computations of isoperimetric functions, aka Dehn functions, which Dehn discovered by working with group invariant polygonal tilings of the hyperbolic plane.

Cocompact Fuchsian groups are also interesting early examples of groups with solvable word and conjugacy problem, using ideas related to Dehn's algorithm.

Cocompact Fuchsian group (together with finite rank free groups) are the early examples of hyperbolic groups in the sense of Gromov, proved using that the group with its word metric is quasi-isometric to the hyperbolic plane.

• More spedifically, Dehn's solution of the word problem for Fuchsian groups generalized itself to small cancellation theory, which is said to have provided a basic set of examples for the development of hyperbolic groups. – ThiKu Jan 24 '17 at 4:42

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.

Are the Fuchsian groups with fixed points interesting from a geometric perspective?


Yes, notice $PSL(2,\mathbb{Z})$ fixes points. The fundamental domain in this case is isometric to a hyperbolic structure on the disk with two cone points.

More generally, if a Fuchsian group $G$ has finite co-volume in $H^2$, then $H^2/G$ will be a hyperbolic 2-orbifold. For example, all of the orientation preserving subgroups of the hyperbolic triangle groups are Fuchsian groups of this type.

In particular, hyperbolic triangle groups have a unique hyperbolic structure, so if 3-manifold has quotient which contains a (orientable) hyperbolic triangle orbifold then the original 3-manifold contains a totally geodesic surface.