Fix some $n\ge 7$. Is there a finite 2-dimensional simplicial complex with
- each vertex is incident to exactly $n$ triangles, and
- the complex forms a closed surface.
If the surface is not closed, then uniform tilings of the hyperbolic plane do the trick.
But I am specifically interested in the case of finite closed complexes.
So is this possible?
Since I want a surface, I made some quick computations with the Euler characteristic $\chi$ of that surface.
I found that the complex must have $-6\chi/(n-6)$ vertices.
I came not much farther than that.
Best Answer
There does always exist such a surface. There is a reasonably short proof using a big tool, namely Selberg's Lemma that every linear group has a torsion free subgroup of finite index.
Assuming equilateral triangles, the symmetry group of the tiling of the hyperbolic plane $\mathbb H^2$ that you depict is a reflection group. In particular, if you pick any tile, then subdivide it into 6 triangles each with angles $\pi/2$, $\pi/3$, $\pi/n$, and then take $T$ to be any one of those 6 triangles, the symmetry group is generated by reflections in the three sides of $T$. That group is a Coxeter group $$C(2,3,n) = \langle a,b,c \mid a^2 = b^2 = c^2 = (ab)^2 = (bc)^3 = (ca)^n = \text{Id} \rangle $$ Furthermore, $T$ is a fundamental domain for the action of $C(2,3,n)$. These conclusions all come from the Poincare Fundamental Polygon Theorem.
We can pass to an index 2 subgroup $R(2,3,n) < C(2,3,n)$ in which the sum of the exponents of $a,b,c$ in each word is even. One can show that $R(2,3,n)$ is generated by the rotations $ab$, $bc$ and $ca$ of order $2,3,n$ respectively, with defining relation $(ab)(bc)(ca) = \text{Id}$. The action of $R(2,3,n)$ on $\mathbb H^2$ therefore preserves orientation. The orientation preserving isometry group of the hyperbolic plane is a linear group, namely $SO(2,1)$. Selberg's Lemma therefore applies, yielding a torsion free finite index subgroup $\Gamma < R(2,3,n) < C(2,3,n)$. The quotient $\mathbb H^2 / \Gamma$ is the surface that you desire.