= Gaussian-integer disc group with an ideal-square Dirichlet domain
{c}
{title2=$D=\{2(|x|+|y|)\le1+x^2+y^2\}\cap\mathbb D$}
Consider determinant-one <matrices> $\left(\begin{smallmatrix}u&v\\\overline v&\overline u\end{smallmatrix}\right)$ over <Gaussian integers> with $u+v\equiv1\pmod{2\mathbb Z[i]}$, modulo their common sign. They act on the <Poincare disc model> by fractional linear maps. Their origin-centered <Dirichlet domain> is
$$
D=\{x+iy: x^2+y^2<1,\ 2(|x|+|y|)\le1+x^2+y^2\}.
$$
For a nonidentity <matrix> set $N=|v|^2\ge1$ and $uv=m+in$. The <group orbit> point is $(m+in)/(N+1)$ and its bisector inequality is $2(mx+ny)\le N(1+x^2+y^2)$. Since $m^2+n^2=N(N+1)$, integrality gives $|m|,|n|\le N$, so the four inequalities defining $D$ imply all of them. They are necessary because the <matrices> with $u=1\pm i$, $v=\pm i$ realize the four combinations $m,n=\pm1$. The four sides are arcs of the unit-radius circles centered at $\pm1\pm i$, and the ideal vertices are $1,i,-1,-i$.
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