Triangulation of hyperbolic surface with boundary

Question

I'm reading Danny Calegari's book scl. On page 10, he gave a proof of the following theorem.

Question Are there any references to the triangulations described here? Is there a proof of the existence of such a "triangulation"?

Let's say $g \geq 2$ and $p = 0$. The standard way I know is to realize $S$ as quotient of a $4g$-gon and cover it by $4g - 2$ triangles. For example,

Lift it to the $m$-fold cover and then the upper bound of the Gromov norm would be $4(g-1)$. So, I'm interested in what kind of "triangulation" the author is using here.

Answer 1

First, a "triangulation" in this sense can be formally defined as a special kind of $n$-dimensional CW complex $X$ for which each open $n$-cell $e \subset X$ has a characteristic map $\chi_e : \Delta^n \to X$ whose domain is the standard $n$-simplex $\Delta^n \subset \mathbb{R}^{n+1}$, such that the following compatibility condition holds:

• For each $k$-dimensional face $\sigma \subset \Delta^n$ there exists an open $k$-cell $e' \subset X$ and a simplicial isomorphism $\psi^{e}_{e'} : \Delta^k \to \sigma$ such that $$\chi_{e'} = \chi_e \circ \psi^{e}_{e'} $$

This is a mild generalization of Hatcher's concept of a $\Delta$-complex structure, which is itself a mild generalization of a simplicial complex structure.

As for Calegari's statement, here is a construction (which I'll describe more intuitively rather than trying to verify the formal details of the above description).

Start with a closed surface $F$ of genus $g$. As you indicated in the diagram in your post, one may triangulate $F$ with $1$ vertex by starting with the triangulated $4g$-gon that you have depicted in the case $g=2$, and then identifying the edges of that polygon in pairs. Of course one must carefully specify a gluing word that has a single vertex cycle, for example this gluing word will do: $$a_1 \, b_1 \, a_1^{-1} \, b_1^{-1} \, ... \, a_g \, b_g \, a_g^{-1} \, b_g^{-1} $$ Inducting on $p$ one gets a triangulation of $F$ with $p$ vertices: assuming one has a triangulation with $p-1$ vertices, pick one triangle $T=\triangle ABC$ and then subdivide $T$ like this (thanks to @MártonPósa for this very nice picture):

Now let's convert a triangulation of $F$ with $p$ vertices into a triangulation of the genus $g$ oriented surface $S$ with $p$ boundary components. Carry out the following alteration one-at-a-time on each of the vertices $V_1,...,V_p$ of $F$:

• pick a triangle $T$ incident to $V_i$;
• pick an embedding of a closed 2-dimensional disc $D \subset T$ which touches the $1$-skeleton only at $V_i$;
• remove the interior of $D$, creating a new boundary component $C_i$ which itself is triangulated with one vertex $V_i$ and with one edge that I will, by abuse of notation, denote $C_i$;
• observe that $T$ has been converted into a 4-gon whose four sides are the original 3 sides of $T$ plus a fourth side $C_i$;
• subdivide that 4-gon into two triangles by inserting one new edge.