I would like to construct a genus $g$ surface regularly tiled by triangles (for example by 238 triangles). Edmunds-Ewing-Kulkarni prove that the only obstruction to doing this is Euler characteristic considerations.

However, I'd like to ensure that the fundamental domain for my tiling contains a ball of some radius. Ideally, I'd like a result of the form: given a radius $R>0$, there is some genus $g_N$ such that for every genus $g \geq g_N$ a surface of genus $g$ can be regularly tiled by triangles using a fundamental domain that contains a ball of radius $R$.

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    What do you mean by large here? Every triangle in $H^2$ has area bounded above by $\pi$, the area of an ideal triangle. However, you should be able to get surfaces with large injectivity radius by combining the facts that surface groups are residually, finite and with appropriate facts about the length spectrum. I think congruence covers of your surface should do the trick. – Neil Hoffman Dec 6 at 18:36
  • I think the following works for even Euler characteristics ($\chi$=2n): take a regular 12n-gon where each edge equals the radius of the inscribed circle. It can be split into 12n triangles. Glue 4n more of such triangles, each to edges k, 4n+k, 8n+k. (This is a generalization of Schmutz's M(2) surface.) – Zeno Rogue Dec 6 at 18:49
  • Sorry, the Euler characteristics is 4-8n, so not every odd genus works; and this is M(3) not M(2). For Euler characteristics 6-12n, a similar generalization of Schmutz's M(4). We also take a 12n-gon, split into 12n triangles, glue another triangle to each edge, and then glue the right edge of triangle number i to the left edge of the triangle number i+5 (numbered clockwise). – Zeno Rogue Dec 6 at 19:38
  • @Neil By large ball, I mean a ball of radius R triangles (measure out R triangles from the center). Congruence subgroups do give a large injectivity radius, which is one way of constructing the type of fundamental domains that I want. However, in general congruence subgroups are rather sparse. I want to show I can do this in every genus larger than some given genus, congruence subgroups will not give every genus – Arielle Leitner Dec 7 at 8:12
  • @Zeno I want the tiling to be preserved (say 238 triangulation) I don't think your construction does this – Arielle Leitner Dec 7 at 8:13

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