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In the hyperbolic-ball model, the limit set of a Kleinian group is the boundary accumulation set of an orbit $Go$ of an interior point: $\Lambda (G)=\overline{Go}\cap \partial {\mathbb{H}}^3$. The choice of $o$ does not matter, since any two such orbits remain a fixed hyperbolic distance apart. It is closed and $G$-invariant. The complicated/non-elementary hypothesis excludes relatively compact groups and elementary groups with at most two limit points; in particular the limit set is nonempty. Having no finite boundary orbit alone would not exclude a compact rotation group, whose interior orbits are bounded.

If $\Lambda (G)$ were finite and nonempty, the orbit of any of its points would be a finite boundary orbit, contrary to the hypothesis. Thus it is infinite. In fact it cannot be countable. We give the essential minimality/perfectness argument. A sequence of Möbius transformations sending an interior point toward $\xi \in \Lambda$ has a subsequence converging on the boundary away from one exceptional point to the constant $\xi$. This follows by normalizing the representing $2\times 2$ matrices: a diverging sequence has a rank-one projective limit, with one kernel line and one image line. For any boundary point $\eta$, the absence of a finite orbit allows choosing $h\in G$ with $h\eta$ different from that exceptional point; then $g_{j}h\eta \to \xi$. Hence the orbit of every $\eta \in \Lambda$ is dense in $\Lambda$.

If $\xi$ were an isolated limit point, density of every orbit would make every point belong to its orbit: each orbit must hit the open singleton $\{\xi \}$. Every point would then be isolated by equivariance, making the compact set $\Lambda$ discrete and therefore finite, a contradiction. Thus $\Lambda$ is a nonempty perfect compact metric space. Repeatedly choosing two disjoint smaller neighbourhoods meeting this perfect set, with diameters tending to zero, associates distinct nested limits to all infinite binary sequences. The limit set is therefore uncountable.

Jorgensen inequality states that for a non-elementary discrete group generated by lifts $A,B\in \mathrm{SL}(2,\mathbb{C})$,For a concrete failure of discreteness, take $A=\mathrm{diag}(e^{s/2},e^{-s/2})$ and $B=\left(\begin{array}{cc}1& 1\\ -1& 0\end{array}\right)$ with $s>0$ small. The fixed points of $A$ and of $BA{B}^{-1}$ collectively include $0,\infty ,-1$, so the group is non-elementary. Direct multiplication gives both summands equal to $4{\sinh }^2(s/2)$. Their sum is less than one when $s$ is sufficiently small, proving this two-generator group is not discrete.

For a connected-limit-set example, the modular group generated by $z\mapsto z+1$ and $z\mapsto -1/z$ is discrete because its matrices have integer entries. It preserves the upper half-plane; its limit set is the extended real line, a circle. Indeed the orbit of the parabolic fixed point infinity consists of the rational points and infinity, dense in that circle. For a disconnected example, choose four pairwise disjoint round disks and two loxodromic maps pairing the exterior of each negative disk with the interior of its positive disk. The Schottky group generated by these maps is free and discrete by the disjoint-domain ping-pong construction. Successive reduced words give nested disjoint disks with shrinking diameters; their limiting points form a Cantor set. Thus its limit set is disconnected, unlike that of the modular group.

Extend the Möbius transformations to isometries of three-dimensional hyperbolic space. For any interior point $o$, the limit set of a Kleinian group is the set of boundary accumulation points of $Go$. Its definition is independent of $o$: two interior orbits stay at a bounded hyperbolic distance, so have the same boundary limits. The Kleinian limit set is closed and $G$-invariant. Iterating a loxodromic Möbius transformation in the positive and negative directions shows that its two fixed points belong to the Kleinian limit set.

The two fixed-point sets in this problem cannot have exactly one common point. Otherwise conjugate that point to infinity and the other fixed point of the first transformation to zero. The transformations then have the forms $f(z)=\alpha z$, with $|\alpha |>1$, and $g(z)=\beta z+b$, with $b\ne 0$. But $f^{-n}g{f}^n(z)=\beta z+b{\alpha }^{-n}$ is a sequence of distinct group elements converging in the Möbius group. Taking quotients of consecutive elements gives nonidentity elements converging to the identity, contrary to discrete subgroup. Consequently the Kleinian limit set contains at least three points.

Let $\xi$ be any point of the Kleinian limit set, and choose distinct $g_n$ with $g_{n}o\to \xi$. The rank-one convergence of divergent Möbius transformations supplies a subsequence converging to $\xi$ uniformly away from one exceptional boundary point $\eta$. To see the lemma directly, normalize determinant-one representing matrices by their norms. The norms tend to infinity, and a subsequential limit has rank one: its kernel is $\eta$ and its image is the attracting point. In the positive-Hermitian-matrix model of hyperbolic space, $g_{n}o$ has this same image limit, so the attracting point is indeed $\xi$.

Choose distinct ${\zeta }_1,{\zeta }_2$ in the Kleinian limit set other than $\eta$. Then $g_n{\zeta }_1$ and $g_n{\zeta }_2$ both tend to $\xi$, belong to the Kleinian limit set, and remain distinct because $g_n$ is injective. If $\xi$ were isolated, both would eventually equal $\xi$, an impossibility. The limit set has no isolated points.

Fix an interior point $o\in {\mathbb{H}}^3$. The limit set of a Kleinian group is the set of boundary accumulation points of its orbit, orThe boundary compactification may be realized as the closed unit ball. A discrete isometry group has only finitely many orbit points in each compact interior region: isometries moving $o$ into a compact set form a compact family after accounting for the compact rotation stabilizer, and infinitely many distinct group elements there would contradict discreteness. The stabilizer of $o$ is finite for the same reason. Thus an infinite group has an infinite orbit escaping every compact interior set and a nonempty boundary accumulation set. A finite group has empty limit set. This also shows the displayed intersection does not introduce ordinary interior accumulation points.

For another point $o^{\prime }$, the corresponding points satisfy $d(\gamma o,\gamma o^{\prime })=d(o,o^{\prime })=:C$. In the Poincaré ball model the distance identity isIf ${\gamma }_{j}o\to \xi \in S_{\infty }^2$, bounded distance impliesSo ${\gamma }_j{o}^{\prime }\to \xi$ as well. Exchanging $o,o^{\prime }$ proves equality of the two accumulation sets. Hence the definition is independent of the interior base point.

It is closed as an intersection of the compact orbit closure with the closed boundary, and it is $\Gamma$-invariant because multiplying an orbit by an element of $\Gamma$ permutes that orbit. Changing ball coordinates or conjugating the geometric realization transports the set by the induced boundary homeomorphism. Different nonconjugate representations of an abstract group are not asserted to have the same subset of one fixed sphere. None of these arguments requires finite generation.

A divergent sequence of Möbius transformations has a subsequence converging uniformly on compact subsets of the Riemann sphere outside one exceptional point to a constant. Represent the transformations by determinant-one $2\times 2$ matrices and divide each by its matrix norm. A convergent subsequence has a nonzero rank-one limit: its determinant tends to zero because the original norms tend to infinity. The kernel line is the exceptional point, and the image line is the constant limit. This converts the limit set of a Kleinian group into a tractable boundary dynamics problem.

A Schottky group is a free, discrete group of Möbius transformations constructed by pairing disjoint Jordan curves on the Riemann sphere, taking the exterior of one bounded disk to the interior of its partner. For round circles it is classical. Strictly separated defining disks give a Cantor set as the limit set of a Kleinian group. The tangent-disk extension permits parabolic elements and is often called a noded or T-Schottky group; its Kleinian limit set need not be a Cantor set.