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The canonical reference, from which all follows, is Marc Troyanov's beautifully written paper: Troyanov, Marc, Les surfaces euclidiennes à singularités coniques. (Euclidean surfaces with cone singula …

Unless I misunderstand the question, the answer is no. The homotopy class of the circle is determined by the partition it determines on the set of marked points, so there are only finitely many homoto …

The thrice punctured sphere has no simple closed geodesics. The four-times punctured sphere has a unique simple geodesic in each homology class. In general, it is a result of I. Rivin that the number …

I don't understand your reference to the model (since the geometry of the hyperbolic plane does not depend on any model), but, in fact, the Beltrami-Klein model demonstrates that any qualitative state …

Isohedral means the tiling has group symmetry, so the quotient is an orbifold, and the smallest area such is the 2-3-7 triangle orbifold, as pointed out by Andre Henriques. However, the $\pi/7$ angle …

I am not sure of the notation, but I assume this can be derived from the Schlafli formula for the volume of a tetrahedron (so this seems to indicate that Gauss knew Schlafli's formula three quarters …

Why do you mean by "what is the form"? And which Gram matrix? (there are two). But if you mean the usual Gram matrix, then the $ij$ element is $-\cosh d(s_i, s_j),$ where $s_i$ is the $2i$-th side of …

The fixed points in the upper half space model map to vectors on the lightcone, which span the plane that intersects with the $\mathcal{I}^3$ at $g$. The details to compute this are here.

Yes, something like that is proved in the paper by Simon Kokkendorff: Kokkendorff, Simon L., Polar duality and the generalized law of sines, J. Geom. 86, No. 1-2, 140-149 (2006). ZBL1115.51010. It w …

I don't really understand the question, perhaps, but if the homeomorphism fixes the boundary, you can extend it by identity to the rest of the surface. This seems to be a homomorphism. Having it be in …

The first reference known to me is Thurston, William P., Three dimensional manifolds, Kleinian groups and hyperbolic geometry, Bull. Am. Math. Soc., New Ser. 6, 357-379 (1982). ZBL0496.57005. Howev …

Each copy $\alpha_i$ of $\alpha$ intersects $\beta$ at a point $x_i.$ Cut $\alpha_i$ at $x_i,$ so you have the top end $t_i$ and the bottom end $b_i$ and connect $b_i$ to $t_{i+1}$ (where $i+1$ is tak …

The question is terribly put , but the answer is: $S_{0, 4}$ is the four times punctured sphere. You can think of this sphere as the ideal simplex in $\mathbb{H}^3$ (it is a theorem of mine that this …

The formula is: $S = \pi,$ thanks to Gauss-Bonnet.

You can gain infinite enlightenment by reading the very cool paper: Gianni, Patrizia; Seppälä, Mika; Silhol, Robert; Trager, Barry, Riemann surfaces, plane algebraic curves and their period matrices, …