Cycle (Fuchsian group)

Description

Imagine a Dirichlet region. Elements of a group move one region to another. Thereby multiple vertices may fall into an orbit where the vertices are acted upon by in a “cyclic manner” by the Fuchsian group.

Definition

Let $R$ be a Dirichlet region of a Fuchsian group $\Gamma$. Two points $p_1,p_2\in R$ are called congruent if there is a $T\in \Gamma$ such that $p_2=T(p_1)$. The equivalence class of congruent points is called a cycle.

The name is taken from the fact that there is one element $T\in \Gamma$ which moves a point too all others in a (finite or infinite) cyclic way.

It is clear, that such points must be vertices of a Dirichlet region or be the mid-points of a geodesic. \

Definition (elliptic and parabolic cycles)

If a cycle is finite it is called elliptic and if the cycle is infinite, it is called parabolic.

Properties

From now on a vertex is an intersection of two geodesic sides or a fixpoint on the boundary located at the middle of a side.

On congruent points and stabiliser elements

Imagine a Dirichlet region $R$ with a vertex $p$. There is a $T\in \Gamma$ moving $R$ to the next region anti-clockwise around $p$. This element needs not to fix $p$. It could just as well map a congruent element $p^{\prime }$ to $p$ and move $p$ somewhere else.

Note, that the angles at congruent points do not need to be the same. I think.

Correspondence to conjugacy classes of cyclic subgrups

Let $R$ be a Dirichlet region. There is a one-to-one correspondence between (elliptic or parabolic) cycles and conjugacy classes of non-trivial maximal cyclic subgroups of $\Gamma$.

Proof: Let $R$ be a Dirichlet region with a corresponding Dirichlet tessellation. Let $p_0,p_1,...,p_n\in H$ be a cycle. Their stabilisers are finitely cyclic subgroups of $\Gamma$ conjugate to one-another. This gives us a map from cycles to a conjugation class. Conversely, every cyclic subgroup of $\Gamma$ has some point $p^{\prime }$ as a stabiliser. This points sits on a vertex of the tessellation and a conjugation moves it to a point in a cycle of $R$. To prove the claim for parabolic cycles, use a point on the boundary of $H$.

The above actually gives us a nice invariant of Fuchsian groups. Just count up all of the orders of the subgroups and obtain an invariant list of numbers. Those numbers are called the Period (Fuchsian group)s.

Earlier I mentioned that the angles of congruent angles do not need to be equal. This seems to be reinforced as there is the following important weaker statement.

The angles of congruent points

Let $F$ be a Dirichlet region for $\Gamma$. Let ${\theta }_1,{\theta }_2,...,{\theta }_n$ be the internal angles at all congruent vertices of $F$. Then ${\theta }_1+...+{\theta }_n=2\pi /m$

Examples

Example