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.


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).


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