Discrete subgroup ($\Gamma \le G$)

A subgroup $G$ of a topological group is discrete when the identity has a neighbourhood containing no other element of $G$. Translation then isolates every group element. In a Hausdorff topological group, such a subgroup is closed: any accumulation at a point would make differences of distinct nearby group elements approach the identity. Consequently its intersection with a compact set is finite. For isometries of hyperbolic space, this converts compactness of the set of isometries moving one point a bounded distance into a properly discontinuous group action.

For example, $(-1/2,1/2)\cap \mathbb{Z}=\{0\}$ isolates the identity element of the additive discrete subgroup $\mathbb{Z}\subset \mathbb{R}$. Its induced subspace topology is the discrete topology, while the surrounding real numbers retain their usual topology; see closed subgroup for why this subgroup is closed but not open.

A discrete additive subgroup of ${\mathbb{R}}^n$ is generated by real-linearly independent vectors. Indeed, isolation of zero gives uniform separation, so compact sets meet the subgroup finitely. Choose a real basis from the subgroup for its real span. Its integer span has finite index, since reduction into a compact fundamental parallelepiped gives finitely many coset representatives. The structure theorem for finitely generated abelian groups then gives a free integer basis, whose size is the dimension of the real span. Thus the subgroup is a Euclidean lattice in its span, although it need not have full rank in the ambient space.

A nonzero discrete additive subgroup of ${\mathbb{R}}^2$ has a basis of one or two real-linearly independent vectors. Choose a shortest nonzero vector $w_1$. If the subgroup is not contained in its line, reduce parallel components modulo $w_1$ and choose a vector $w_2$ of least positive perpendicular height. Such a height exists: heights tending to zero would give infinitely many reduced points in a bounded region, contrary to discreteness. Reducing any group vector modulo $w_2$ leaves height smaller than this minimum, hence height zero, and reducing along $w_1$ then gives an integer multiple of $w_1$.

A discrete subgroup of ${\mathrm{PSL}}_2(\mathbb{C})$, acting by Möbius transformations on the Riemann sphere or orientation-preserving isometries of hyperbolic space ${\mathbb{H}}^3$. A group may be Kleinian even when some elements fix points; freeness and absence of torsion are additional conditions.

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 a non-elementary discrete subgroup generated by $A,B\in \mathrm{SL}(2,\mathbb{C})$, the displayed inequality is a necessary trace constraint. Failure of it proves nondiscreteness when non-elementarity has independently been checked. It is not a sufficient condition for discreteness and must not be applied to an elementary pair without its hypothesis.

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.

A Tangent-disk Schottky group uses side-pairing disks with disjoint interiors but permits boundaries to meet. This convention must be distinguished from the strictly separated, purely loxodromic Schottky group convention. For example, the maps $A(z)=z+2$ and $B(z)=z/(iz+1)$ pair the half-plane disks $\Re z<-1,\Re z>1$ and the disks $|z-i|<1,|z+i|<1$. The ping-pong lemma gives a free discrete group. With $C=AB{A}^{-1}{B}^{-1}$, the subgroup generated by $B,C$ is conjugate by $M(z)=iz/(2-z)$ to the principal level-two modular group, generated by $\left(\begin{array}{cc}1& 0\\ 2& 1\end{array}\right)$ and $\left(\begin{array}{cc}1& 2\\ 0& 1\end{array}\right)$. Thus its Kleinian limit set contains the circle $|z-1|=1$ and its translate $|z-3|=1$. Its Hausdorff dimension is at least one and it cannot be homeomorphic to a single circle.

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 nontrivial finite-order elliptic Möbius transformation fixes an axis in hyperbolic space. The quotient metric has cone angle $2\pi /m$ along that axis for a stabilizer of order $m$, so it is locally an orbifold metric rather than a smooth hyperbolic-manifold metric. The underlying topological quotient may nevertheless be a manifold.

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.

For a discrete isometry group of hyperbolic space, its limit set is the boundary accumulation set of one interior orbit. All interior base points give the same set: corresponding orbit points stay a fixed hyperbolic distance apart, and bounded-distance sequences approaching the boundary have the same endpoint. This definition does not require finite generation.

Every nonempty closed boundary subset invariant under a non-elementary Kleinian group contains the Kleinian limit set. Such a subset cannot be a singleton. Its hyperbolic convex hull contains an interior point, whose orbit lies in the hull; in the Klein ball model the boundary of this hull is contained in the original closed boundary subset.

The Kleinian limit set of an infinite subgroup is nonempty. If it is normal, its limit set is invariant under the ambient group by base-point independence and conjugation. Minimality of the Kleinian limit set gives one inclusion, and subgroup containment gives the other. Neither properness nor finite generation of the subgroup is needed.

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.

The integer matrices $A=(\begin{array}{cc}1& 2\\ 0& 1\end{array})$ and $B=(\begin{array}{cc}1& 0\\ 2& 1\end{array})$ generate a discrete subgroup. For every nonzero integer $m$, $A^m$ sends $|x|<1$ into $|x|>1$, while $B^m$ sends $|x|>1$, including infinity, into $|x|<1$. The ping-pong lemma identifies their subgroup as the free product of two infinite cyclic groups. This separates the proof of discreteness from the proof of freeness.