Let $p_k := e^{\pi/2 i k}$, $k \in \{0, 1,2,3\}$. Let $b_k$ the geodesic of the hyperbolic disk connecting $p_k$ and $p_{k+1(\text{mod}4)}$. For instance, $p_0$ and $p_1$ are connected by the lower left quarter-circle of center $(1+i)$ and radius 1. Let $\alpha_k$ be the element in $\text{Aut}(\mathbb{D})$ that ''reflects'' along the geodesic $b_k$. Let $G < \text{Aut}(\mathbb{D})$ be the subgroup generated by the $\alpha$'s. Is $G$ a Fuchsian Group?
Moreover, let $\beta_1 \in \text{Aut}(\mathbb{D})$ be the orientation-preserving transformation that sends $b_0$ to $b_1$ (keeping $p_1$ fixed) and $\beta_2 \in \text{Aut}(\mathbb{D})$ the (orientation-preserving) one that sends $b_2$ onto $b_3$ (keeping $p_3$ fixed). Define $H$ to be the group generated by the $\beta$'s. Is it a Fuchsian group?
I would be very grateful if someone could give me a ''graphical'' explanation of how the ''typical'' Fuchsian group looks like.
$H$ is a Fuchsian group in the classical sense. The main properties are:
The group $G$ contains reflections, so it depends on the broader understanding of Fuchsian groups.
Since you asked for something “graphical”, here are your two groups visualized as subgroups of a given triangle reflection group, with triangles of the same orbit colored in the same color. The corners don't form the ideal square you described, but that's just a global Möbius transformation missing here, so no “real” difference.