From an octagon to a genus 2 surface - Mathlapse

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Recommends Lee Mosher · 2019-12 · Mathematics Stack Exchange · on “Example of cocompact discrete subgroups of $\operatorname{SL}_2(\mathbb R)$”

Many examples arise by application of the Poincare polygon theorem, which you can find for example in Theorem 11.2.1 of Ratcliffe's book "Foundations of Hyperbolic Manifolds". Here's a general description. Consider a finite polygon $P$ with $2n$ sides. Let $f_1,...,f_n \in \text{Isom}(\mathbb H)$ be isometries satisfying the following conditions: For each $i=1,...,n$ there exist sides $a_i,a'_i$ such that $$P \cap f_i(P) = a_i = f_i(a'_i) $$ The list of sides of $P$ is $a_1,a'_1,...,a_n,a'_n$ . The total angle of each "vertex cycle" is $2\pi$ (more can be said here, but I'll refer to topology textbooks that cover the classification of surfaces). The conclusion of the Poincare Polygon Theorem

Cites Hamish Todd · 2017-10 · Mathematics Stack Exchange · on “Gluing an octagon up into a genus 2 surface in a highly symmetrical way”

This video [this video] is a representation of the octagon-to-genus-2-surface process. But there's a more symmetrical representation seen here: [LINK] The surface we see in the above picture isn't as symmetrical as it could be though (and the way it is glued up isn't as simple and symmetrical as in the video above either). I want to make a nice video + animation about genus 2 surfaces (and their connection to the hyperbolic plane). The maximal symmetry on a genus 2 surface in (in euclidean 3 space) is like one of these: [LINK] [LINK] Can anyone think of a simple, ideally highly symmetrical way of constructing either of those from the octagon at the top there?

Cites user211599 · 2018-05 · Mathematics Stack Exchange · on “Given a fundamental polygon and an attaching word, is the resulting space "unique?"”

There are at least two ways of gluing an octagon to form a double torus. One is displayed in this YT video [this video] . There is another way by identifying opposite sides. Here is an image detailing the process. [LINK] Now, this construction is intrinsically different from the one linked in the video. The $cd^{-1}c^{-1}d$ path in the image is a loop the is homologous to zero, yet not null-homotopic. This is not true for the two pairs of opposite sides in the video. A sequence of edges on the fundamental polygon can correspond to a loop on the resulting surface. For example, in the image, $adc^{-1}$ corresponds to a loop going around a hole in the double torus. It is my suspicion that these loop

Cites wilsonw · 2019-02 · Mathematics Stack Exchange · on “Fundamental group of a genus-2 surface”

Am I right to say that since a genus-2 surface can be glued from an octagon (see, e.g. [this video] ) and so reading from its boundary, $[a_1,b_1][a_2,b_2]$ can be retracted to a point, and thus the fundamental group as given above?

Cites Semiclassical · 2023-05 · Mathematics Stack Exchange · on “Joining the edges of a disc in a way that prevents mirroring”

If we replace the disk with a $2n$-gon, what you're describing sounds similar to the construction of a surface with genus $g$ by gluing (see here [this video] for an animated example of genus 2). This falls under the domain of algebraic topology to my understanding. Surfaces of infinite genus have been considered, but they can be much wilder (see for instance the Loch Ness monster surface [LINK] .)