# Reference for 'Normal Subgroups of Fuchsian Groups'

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

• This question is far too vague as stated. I'm voting to close. (Though the two answers are the best one might hope for.) – HJRW Jan 16 '17 at 17:13

## 2 Answers

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".

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.