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The Gaussian Bianchi group is the Kleinian group of determinant-one two-by-two matrices over the Gaussian integers, modulo $\{\pm I\}$. Its matrix entries lie in a discrete subring of $\mathbb{C}$, so it is a discrete subgroup of $PS{L}_2(\mathbb{C})$. The Borromean link group is a torsion-free subgroup of index 24.

For $n\in \mathbb{Z}[i]$ and a nonzero integer $r$, the displayed determinant-one matrix has trace two and represents a nonidentity parabolic Möbius transformation fixing $n/r$. Its fixed-point equation is $(rz-n{)}^2=0$. These parabolic fixed points are dense in the Riemann sphere, proving that the Gaussian-integer determinant-one Kleinian group has full limit set.

A complete hyperbolic three-manifold is a complete Riemannian manifold of dimension three with constant sectional curvature $-1$. Its universal cover is hyperbolic space ${\mathbb{H}}^3$ and its deck group acts freely and discretely by Riemannian isometries. In the orientable case it is a torsion-free Kleinian group.

A Kleinian group whose Kleinian limit set contains at least three points. A discrete group fixing a common boundary point has at most two limit points and is therefore elementary.

Suppose a connected subset $C\subset F$ contains distinct points $a,b$. The distance function $x\mapsto |x-a|$ is $1$-Lipschitz, and its image on $C$ is connected and contains $0$ and $|b-a|$. It therefore contains the whole interval between those values. A Lipschitz map cannot increase Hausdorff dimension: taking images of arbitrarily fine covers multiplies their diameter powers by at most a fixed constant. Since a nondegenerate interval has Hausdorff dimension one, ${\dim }_{H}C\ge 1$, contradicting ${\dim }_{H}F<1$. Thus every connected subset is a singleton, so $F$ is totally disconnected.

A Möbius transformation pairs two disks when it maps their boundary circles to one another and takes the exterior of the first disk onto the interior of the second. Two pairs of disks with disjoint interiors give the ping-pong lemma construction of a free two-generator group. Tangencies are allowed in the convention suggested by the question; this is often called a Tangent-disk Schottky group, rather than a strictly separated Schottky group.

Here is a concrete configuration that also makes the dimension argument precise. Take $A(z)=z+2$ and $B(z)=z/(iz+1)$. The first pairs the half-plane disks $\Re z<-1$ and $\Re z>1$, which are disks on the sphere tangent at infinity. The second pairs $|z-i|<1$ and $|z+i|<1$, tangent at zero; their interiors are also disjoint from the half-plane disks. These pairings give a free discrete Kleinian group by ping-pong lemma, with a nonempty open fundamental region.

Set $C=AB{A}^{-1}{B}^{-1}$. Conjugating the subgroup generated by $B,C$ by $M(z)=iz/(2-z)$ gives matricesThe product of the second matrix with the first is $\left(\begin{array}{cc}1& 2\\ 0& 1\end{array}\right)$. These are the standard parabolic generators of the principal level-two modular group, whose ideal-quadrilateral fundamental region tiles the upper half-plane and whose Kleinian limit set is the entire extended real line. Consequently the original group's Kleinian limit set contains the circle $|z-1|=1$, as well as its image under $A$, the distinct tangent circle $|z-3|=1$.

It follows that the limit set has Hausdorff dimension at least one and is not homeomorphic to a circle. Indeed a proper subset of a circle cannot itself be a circle: after deleting a missing point it lies in a line, which cannot contain an embedded circle. A space containing two distinct complete circles thus cannot be homeomorphic to a single circle. This explains why dimension at least one does not force circle topology. The tangent-circle terminology is consistent with the primary discussion of T-Schottky configurations in arxiv.org/pdf/math/0701572 .

A Kleinian group is a discrete subgroup of the orientation-preserving isometry group of ${\mathbb{H}}^3$, identified with ${\mathrm{PSL}}_2(\mathbb{C})$. For an abstract finitely generated group this means a faithful representation into ${\mathrm{PSL}}_2(\mathbb{C})$ whose image is discrete; the chosen image is the geometric realization. Its boundary action is by Möbius transformations on $S_{\infty }^2\cong \mathbb{C}\cup \{\infty \}$.

Discreteness means the identity is isolated in the group topology, and for a subgroup it also excludes an infinite sequence of distinct elements converging anywhere in the isometry group. Torsion-freeness is not part of this definition: elliptic Möbius transformations of finite order are allowed. Therefore a Kleinian quotient need not be a smooth hyperbolic manifold.

Write $N={\Gamma }^{\prime }$. As a subgroup of a discrete group, it is a Kleinian group, even if it is not finitely generated. Since it is infinite, part (b) gives a nonempty closed Kleinian limit set $\Lambda (N)$. For $\gamma \in \Gamma$, conjugation and base-point independence giveOne can see this directly by applying $\gamma$ to $No$: the image is the orbit of $\gamma o$ under the conjugate subgroup, and the choice of interior point does not affect its limit set. Normality makes the conjugate subgroup equal to $N$, so $\Lambda (N)$ is $\Gamma$-invariant. Applying part (c) yields $\Lambda (\Gamma )\subseteq \Lambda (N)$.

Conversely $No\subseteq \Gamma o$, so every boundary accumulation point of the smaller orbit is one of the larger orbit: $\Lambda (N)\subseteq \Lambda (\Gamma )$. Therefore the limit set of an infinite normal Kleinian subgroup satisfiesInfinitude supplies nonemptiness, normality supplies invariance, and non-elementarity supplies minimality. Properness of the subgroup is not needed, and no finite-generation assumption on it has been inserted.

A Kleinian group is a discrete subgroup of ${\mathrm{PSL}}_2(\mathbb{C})$, acting as orientation-preserving hyperbolic isometries of ${\mathbb{H}}^3$ and as Möbius transformations of its boundary sphere. Its limit set consists of the accumulation points on the boundary of an orbit $Gx$ with $x\in {\mathbb{H}}^3$; it is independent of that interior base point and is closed. A nonidentity parabolic transformation is conjugate to $z\mapsto z+c$, $c\ne 0$. Its iterates on the upper half-space tend to its unique fixed boundary point, infinity in these coordinates. Consequently every parabolic fixed point belongs to the limit set.

The displayed matrix has determinant one, trace two, and is not the identity because $a\ne 0$. Its fixed-point equation is $a(z-w{)}^2=0$, so it has exactly the double fixed point $w$. It is therefore parabolic, and its fixed point is $w$.

The determinant-one Gaussian-integer matrices form a subgroup of ${\mathrm{SL}}_2(\mathbb{C})$. They are discrete because their entries lie in a discrete lattice: a matrix sufficiently near the identity has every entry equal to the corresponding identity entry. Passing through the quotient by $\{\pm I\}$ preserves discreteness, so their Möbius transformations form a Kleinian group.

For $w=n/r$, $n=p+iq$ and $r\ne 0$, set $a=r^2$. This gives the Gaussian-integer parabolic constructionAll entries lie in $\mathbb{Z}[i]$, the determinant is one, and its parabolic fixed point is $n/r=w$. The points with nonzero $p,q,r$ are dense in the finite complex plane, and their closure on the sphere includes infinity. Since the limit set is closed and contains all these parabolic fixed points, the limit set is the whole Riemann sphere.

A Kleinian group is a discrete subgroup of ${\mathrm{PSL}}_2(\mathbb{C})$, acting by Möbius transformations on the Riemann sphere, equivalently by orientation-preserving isometries of hyperbolic space ${\mathbb{H}}^3$.

Consider the classes ofThey generate a discrete subgroup because their matrix representatives have integer entries: a sequence of such matrices converging to $I$ or $-I$ is eventually constant. On the real projective line set $X=\{x:|x|>1\}\cup \{\infty \}$ and $Y=\{x:|x|<1\}$. For every nonzero integer $m$,The last inclusion follows from $|2mx+1|\ge 2|x|-1>|x|$ when $|x|>1$; also $B^m(\infty )=1/(2m)\in Y$. The ping-pong lemma therefore says that the subgroup is the free product $⟨A⟩\ast ⟨B⟩$. Both generators have infinite order, soa free group on two generators. The ping-pong inclusions ensure that no nonempty reduced word can act as the identity, which is the essential freeness argument.