Reference for 'Normal Subgroups of Fuchsian Groups'

Question

I am looking for a reference on how to explicitly construct normal subgroups of a given Fuchsian group. I appreciate any help.

Answer 1

Fuchsian groups are residually finite, so they have many finite index normal subgroups. (Even better, surface groups are residually free!)

Torsion free fuchsian groups (aka surface groups) are even nicer - you can use homology arguments to produce normal subgroups of various kinds. Surface groups also have a wild variety of infinitely generated infinite index normal subgroups.

If you can say a bit more about what kind of normal subgroups you need, perhaps somebody could give a more useful answer.

Finally - since you ask for a reference - here is a pleasantly short discussion of surface groups, with a pointer to Stillwell's lovely introductory book "Classical topology and combinatorial group theory".

Answer 2

Given a specific example of a finitely presented group $G$ and a fixed $n$, there are well-known methods for enumerating the normal finite index subgroups $H$ such that $[G:H]=n$. One such method is Holt's homomorphism algorithm.

Essentially, one can choose a finite group $K$ of order $n$ and test to see if $G$ admits a surjective homomorphism $f$ onto $K$ by mapping the generators of $G$ onto a set of elements $S_K$ $K$ and then testing to see that 1) the relators of $G$ map to the identity in $K$ and 2) that map is surjective. However, it is prudent to only consider the equivalence classes of $S_K$ under automorphisms of $K$. And then the kernel of $f$ is given as a normal subgroup of $G$.

This streamlining has been implemented in existing software: GAP and MAGMA (although MAGMA is subscription based it is licensed to all North American Universities via a grant from the Simons Foundation).

This method of enumerating finite index normal subgroups sketched above is Chapter 9 of:

Holt, D. F., Eick, B., & O'Brien, E. A. (2005).
Handbook of computational group theory. CRC Press.

However, if one only wants to construct some normal subgroups of a finitely generated Fuchsian group. Then one can compute a discrete faithful representation of a Fuchsian group $\rho:G \rightarrow PSL(2,\mathbb{R})$ such that all of the entries of elements are algebraic numbers. The entries of the generators define a ring of $S$-integers $\mathcal{O_S}$ (possibly a ring of integers) in some number field with a real embedding. Consider a prime ideal $\mathcal{J}$ of $\mathcal{O_S}$, then there exists a homomorphism from $G$ into $PSL(2,O_S/\mathcal{J})$ given by reduction mod $\mathcal{J}$ of each entry of the representation. Given a fixed a ring of $S$-integers $\mathcal{O_S}$ there will be infinitely many such $\mathcal{J}$, each normal subgroup of $G$ will be given by $\rho(G) \cap PSL(2,\mathcal{J})$.

Notice, we can slightly relax the assumption that $G$ is finitely generated if we can find $\rho:G \rightarrow PSL(2,\mathbb{R})$ where all of the entries of the generators lie in ring of $S-$integers of some number field.