Timeline for Examples of elementary group of isometries of the ideal boundary of hyperbolic plane
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| Mar 27, 2023 at 18:42 | comment | added | Robbie Lyman | More to the point, if $\Sigma$ is a finite-area hyperbolic surface (without boundary), which we identify with $\Gamma\backslash\mathbb{H}^2$ for some discrete, torsion-free subgroup $\Gamma \le \operatorname{Isom}(\mathbb{H}^2)$, then $\Gamma$ is never elementary. I won't attempt to prove this. | |
| Mar 27, 2023 at 18:37 | comment | added | Robbie Lyman | I suppose the prime examples fixing the point at infinity in the upper half-plane model of $\mathbb{H}^2$ are the infinite cyclic groups generated by the horocyclic translation $z \mapsto z + 1$ (should fall under definition 2, but I'd have to give Busemann functions a proper think to say for sure) and the loxodromic isometry $z \mapsto 2z$, which preserves the geodesic line $\Re z = 0$. In both cases, the quotient surface is topologically an open annulus and has infinite area; in the former it has one cusp. | |
| Mar 27, 2023 at 18:05 | history | asked | Quanta | CC BY-SA 4.0 |