# Mapping Class Group and Triangulations

I am a physicist who's getting started with Mapping Class Group for Riemann surfaces, pants decompositions and triangulations so I apologise in advance if the following is a stupid question/wrong.

I understand that to any pants decomposition of a Riemann surface we can associate a set of generators (the Dehn twists). Different pants decompositions gives different sets of generators, and relations among various sets of generators are understood as being generated by a minimal sets of relations (Lantern, Chain, Braiding...)

My question is: is there a similar picture for Triangulations? Given a Triangulation, can I assign a canonical set of generators of the Mapping Class Group? Can I understand relations between generators using flips of triangulations?

Yes. When a group $$G$$ acts geometrically on a metric space $$X$$, by choosing a basepoint $$x_0 \in X$$ you can construct its Dirichlet domain $$D_{x_0} = \{x \; | \; d(x, x_0) \leq d(x, g \cdot x_0) \; \forall g \in G\}$$ When the action of $$G$$ is sufficiently nice, this domain has finitely many sides and geodesics which are perpendicularly bisected by each face form a finite generating set for $$G$$.

Since the mapping class group acts geometrically on the (labelled) flip graph (with the graph metric) we can do a similar process starting at a triangulation $$\mathcal{T}_0$$.

1. Let $$X_1$$ be the set of mapping classes which move $$\mathcal{T}_0$$ by the smallest non-zero amount.
2. Let $$X_2$$ be the set of mapping classes which move $$\langle X_1 \rangle \cdot \mathcal{T}_0$$ by the smallest non-zero amount.
3. Let $$X_3$$ be the set of mapping classes which move $$\langle X_1, X_2 \rangle \cdot \mathcal{T}_0$$ by the smallest non-zero amount.
4. Let $$X_4$$ be the set of mapping classes which move $$\langle X_1, X_2, X_3 \rangle \cdot \mathcal{T}_0$$ by the smallest non-zero amount.

$$\vdots$$

Then each $$X_i$$ is finite and for some $$N$$ the elements of $$X_1 \cup X_2 \cup \cdots \cup X_N$$ generate $$G$$. Since each of generator $$g$$ can be represented by a path in the flip graph from $$\mathcal{T}_0$$ to $$g(\mathcal{T}_0)$$ the relations between these generators can then be understood from the 2--cells of the flip graphs which give:

• the square relation - that disjoint flips commute, and
• the pentagon relation - that two flips which share a common triangle form a 5--cycle

Since there are explicit descriptions of the action of the mapping class group on the flip graph this entire process can be done on a computer (although as far as I am aware no one has actually done this).