# Torsion-free, normal subgroups of certain Coxeter groups

Let $G$ be the reflection group of a regular, 4-dimensional, hyperbolic honeycomb. I would like to find a family $H_i < G$ of finite-index, torsion-free subgroups of $G$, so that I can represent the quotients $G/H_i$ in some computer algebra system like GAP or Magma.

I already tried to do this by defining $G$ as a finitely-presented group in Magma and using its "LowIndexNormalSubgroup"-method. However, the index becomes too large for Magma to handle.

A different approach I came up with is using a faithful representation of $G$ in some $PSL_m(\mathbb{Z}[ \xi_1,...,\xi_n ])$ and then factoring out a family of suitable family of ideals of $\mathbb{Z}[ \xi_1,...,\xi_n ]$. This has been done for 2D in

Mennicke J. Eine Bemerkung über Fuchssche Gruppen. Inventiones mathematicae. 1967 Aug 1;2(4):301-5.

Given that I am far from being an expert in this subject I would like to know whether someone here could help by pointing me in the right direction.

## 1 Answer

There has been quite a bit of activity in "abstract regular polytopes", i.e. certain kinds of quotients of Coxeter groups with string diagram. See e.g. the book by P.McMullen and E.Schulte "Abstract regular polytopes", CUP, 2002. One way to construct such examples is to factor out an invariant sublattice (of finite index) in the natural reflection representation of your group. Indeed, there are constructions giving you finite examples, already often it suffices to work in something like (citing from memory) $SL_4(\mathbb{Z}[\sqrt{2}])$.

• Thank you for your response. I will have a look at the reference you gave. Regarding the construction; the representation in Mennickes paper kind of falls from the sky. I assume he got it by considering the Klein-model of $\mathbb{H}^2$ and working out the reflections there. What is mysterious to me is how does one enforce getting only integers in the representation. Follow up question: why do we adjoin $\sqrt{2}$ in your example? Is there a recipe for finding these representations? Thank you! – Nikolas Breuckmann Apr 12 '16 at 22:27
• it depends; if I recall correctly, $\sqrt{2}$ appears when you work with crystallographic groups. You will find all the details in the book I cited, or in one of their papers. Here is another work of this kind : sciencedirect.com/science/article/pii/S019688580400003X – Dima Pasechnik Apr 12 '16 at 22:56
• The book and the references therein were very helpful. Thanks again. – Nikolas Breuckmann Apr 18 '16 at 12:13