Finite-volume hyperbolic three-manifolds have finite standard cusp decompositions

We prove the needed uniform small-matrix fact. In a neighborhood of the identity of ${\mathrm{P}\mathrm{S}\mathrm{L}}_2(\mathbb{C})$ choose the unique determinant-one lift near $I$, and put $\rho (g)=\max \{\mathrm{\parallel }g-I\mathrm{\parallel },\mathrm{\parallel }{g}^{-1}-I\mathrm{\parallel }\}$ for a fixed matrix operator norm. For small $r$, the symmetric neighborhood $U_r=\{\rho <r\}$ has compact closure. Writing $g=I+X,h=I+Y$ gives $[g,h]-I=(XY-YX)g^{-1}{h}^{-1}$. The same formula with $g,h$ reversed bounds the inverse commutator, so $\rho ([g,h])\le C\rho (g)\rho (h)$ on $U_r$, for one fixed $C$. Choose $r$ with $Cr\le 1\mathrm{/}2$. For any discrete subgroup $D$, the finite set $S=D\cap U_r$ generates a nilpotent subgroup. Indeed, every left-nested commutator of $n$ elements of $S$ has $\rho \le r{2}^{-(n-1)}$, by induction, and remains in $U_r$. Since the identity is isolated in $D$, all such commutators vanish for one common sufficiently large $n$. To justify the nilpotence conclusion, argue by induction on this common bound: the commutators of length $n-1$ commute with every generator, hence generate a central subgroup; quotient by it and use the induction hypothesis. The initial bound $n=1$ is the trivial group. Thus the group has a finite central series. Compactness makes $D\cap U_r$ finite, since a discrete subgroup is closed as in [F2]'s elementary proof.

We pass from small matrices to small point displacement, including the finite-index step. Fix a model point $o$ and a compact neighborhood $Q$ of its compact stabilizer $K$. Choose an identity neighborhood $V$ so that $qV{q}^{-1}\subset U_r$ for every $q\in Q$, possible by uniform continuity on compact $Q$. Cover $K$ by finitely many open sets $W_1,\dots ,W_N$ with $W_j^{-1}{W}_j\subset V$. There is ${\eta }_0>0$ such that every $g$ with $d(o,go)<{\eta }_0$ lies in $Q\cap {\bigcup }_j{W}_j$. To verify this last assertion, the inverse image of the closed unit ball under the frame projection $G\to {\mathbb{H}}^3$ is compact: use finitely many local trivializations over that ball with compact fiber $K$. A contrary sequence with displacement tending to zero therefore has a subsequence converging into $K$, a contradiction. Choose $0<\eta <{\eta }_0\mathrm{/}(N+1)$. For a discrete torsion-free subgroup $\mathrm{\Gamma }$ let $D$ be generated by the elements moving $o$ by less than $\eta$; there are finitely many such elements, again by the compact frame fiber. Set $H=⟨D\cap U_r⟩$, nilpotent by step 1.1. In any word of length greater than $N$, its first $N+1$ prefixes lie in $Q\cap {\bigcup }_j{W}_j$, by the triangle inequality. Two prefixes $w_i,w_j$, $i<j$, lie in the same $W_j$, so $h=w_i^{-1}{w}_j\in V\cap D$ and $w_{i}h{w}_i^{-1}\in U_r\cap D\subset H$. If the rest of the word is $v$, then $w_{j}v=(w_{i}h{w}_i^{-1})w_{i}v$. Deleting this positive-length block therefore gives a shorter representative of the same left coset modulo $H$. Iteration shows every left coset has a representative of length at most $N$, proving $[D:H]<\mathrm{\infty }$.

Such a $D$ preserves either a unique ideal point or a pair of ideal points, unless trivial. If $H$ is trivial, finite index and torsion-freeness make $D$ trivial. Otherwise its normal core in $D$ has finite index and is nontrivial and nilpotent. A nontrivial nilpotent group has a nontrivial center, by its central series; choose a central element $z$. Its centralizer from [F2] puts that normal core into either the translations fixing the parabolic point of $z$ or the diagonal group fixing its loxodromic pair. It is abelian and has that unique common fixed set. Normality makes $D$ preserve that set. A discrete torsion-free group preserving a pair is cyclic: an element interchanging the pair has order two, and the remaining diagonal group is cyclic by [F2]. A group preserving one point consists either of translations or is cyclic loxodromic. For the latter assertion, conjugate a loxodromic element fixing that point to a dilation fixing infinity and zero. Its commutator with any affine map fixing infinity is a translation. A nonzero such translation, conjugated by contracting powers of the dilation, tends to the identity, impossible in a discrete group. Thus all affine maps commute with the dilation and lie in the cyclic diagonal group. This proves the assertion and also that $D$ is abelian. By model homogeneity, the same $\eta$ works at every point.

Consider the open displacement regions $U_g=\{x:d(x,gx)<\eta \}$ for nonidentity $g\in \mathrm{\Gamma }$. If two meet, step 3.1 applied at an intersection point says that their elements have the same parabolic fixed point or the same loxodromic pair. This propagates along every intersection chain. Each $U_g$ is connected: for a translation it is a horoball, since $\sinh (d((z,y),(z+\lambda ,y))\mathrm{/}2)=\mathrm{\mid }\lambda \mathrm{\mid }\mathrm{/}(2y)$; for a loxodromic with translation length $\mathrm{\ell }>0$ and rotation angle $\alpha$, cylindrical coordinates about its axis give ${\sinh }^2(d(x,gx)\mathrm{/}2)={\sinh }^2(\mathrm{\ell }\mathrm{/}2){\cosh }^{2}r+{\sin }^2(\alpha \mathrm{/}2){\sinh }^{2}r$, so it is an axis tube or empty. This formula follows from the hyperboloid coordinates $(\cosh r\cosh t,\cosh r\sinh t,\sinh r\cos \phi ,\sinh r\sin \phi )$ and the shift $(t,\phi )\mapsto (t+\mathrm{\ell },\phi +\alpha )$. Thus every component of ${\bigcup }_g{U}_g$ is associated to exactly one fixed point or pair. Its stabilizer is the full elementary subgroup with that fixed set, by step 3.1. It is precisely invariant under this stabilizer: an intersecting translate gives an intersection chain and hence the same fixed set.

A parabolic component is one horoball: its stabilizer is a nontrivial discrete translation lattice of rank one or two, with a shortest nonzero translation vector, and the union of the corresponding concentric horoballs is the horoball for that shortest vector. A loxodromic component is an axis tube of finite radius. Its stabilizer is generated by one screw motion with positive translation length $\mathrm{\ell }$. Only finitely many of its powers satisfy $\mathrm{\mid }n\mathrm{\mid }\mathrm{\ell }<\eta$; the formula in step 4.1 gives one finite tube radius for each such power, and their union has the largest of these finite radii. The quotient of its closed tube is compact, since the angular coordinate is a circle, the axis coordinate is taken modulo $\mathrm{\ell }$ with a possible rotation, and the radial coordinate lies in a bounded closed interval. Thus the thin open subset of $M$ has precisely the claimed tube and horocusp component models. Its complement is the closed set on which injectivity radius is at least $\eta \mathrm{/}2$, since injectivity radius in a simply connected hyperbolic cover is half the infimum of the nontrivial deck displacements.

This thick complement is compact. Otherwise properness [F3] lets us choose infinitely many points there mutually separated by at least $\eta \mathrm{/}4$: successively avoid finitely many compact closed balls. Their radius-$\eta \mathrm{/}8$ balls are disjoint and are isometric embedded hyperbolic balls, since the injectivity radius at each center is at least $\eta \mathrm{/}2$. Each has the same positive volume, contradicting finite total volume. The thick set is nonempty. If empty, connectedness of $M$ puts its entire cover into one component of the thin set from step 4.1; $\mathrm{\Gamma }$ is then elementary. A translation rank-two quotient has infinite volume by integrating $y^{-3}$ down to zero over a fundamental parallelogram; rank-one quotients have an infinite transverse horizontal direction. A cyclic loxodromic quotient has infinite volume by integrating $\sinh r\cosh rdrd\phi dt$ over unbounded $r$ and an axis fundamental interval. The trivial quotient is also infinite-volume. All contradict the hypothesis.

Only finitely many thin components occur. To prove the required local finiteness near the compact thick set, choose a small lifted metric ball about any one of its points. Every displacement region $U_g$ meeting a radius-$a$ ball about its center $x$ has $d(x,gx)<\eta +2a$ by the triangle inequality. There are finitely many such $g$, because the inverse image of a bounded displacement ball in the model frame bundle is compact and its intersection with the discrete group is finite. Each $U_g$ belongs to a single component, so only finitely many components meet this neighborhood. Finitely many such neighborhoods cover the thick set. Every thin component has boundary in that set, since it is a proper component of the open thin locus and the displacement descriptions in steps 4.1–5.1 supply a boundary. Therefore every component occurs in that finite list.

A rank-one horocusp has infinite volume: a translation fundamental strip has an infinite transverse Euclidean direction, and integrating $y^{-3}$ over any finite nonempty height interval already gives infinite volume. It is excluded. The remaining finitely many horocusps are rank two and have standard compact torus cross-sections. We justify the closed boundaries before taking their complement. Choose $\eta <{\eta }^{\mathrm{\prime }}<{\eta }_0\mathrm{/}(N+1)$ and rerun steps 2.1–3.1 at ${\eta }^{\mathrm{\prime }}$; their finite-index and elementary-group conclusions still hold. At a common point of two closed $\eta$-displacement regions the two generators move that point by at most $\eta <{\eta }^{\mathrm{\prime }}$, so they have the same elementary fixed set. Consequently distinct closed thin components cannot touch, and each closure is precisely invariant under its stabilizer. The closed tube quotients embed because they are compact and injective; the closed rank-two horoball quotients embed properly by [F2], whose explicit properness proof applies to every precisely invariant closed horoball with compact torus cross-section. Their boundaries are thus disjoint embedded smooth model tori. Add the finitely many compact tube closures from step 5.1 to the compact thick set from step 6.1. This is exactly the complement of those open rank-two horocusps, hence compact. Each cusp boundary is a smooth horotorus by its explicit model, and its attached product end is the horoball quotient. Clamping each product end to its boundary gives a deformation retraction onto the complement, so this is a compact core in the homotopy sense as well. This constructs the claimed finite standard cusp decomposition, including the zero-cusp compact case.