Poincaré's 1883 tangent conjecture for the limit curve of second-family Kleinian groups (necklace groups)

In his 1883 memoir on Kleinian groups, Henri Poincaré constructed the second family of Kleinian groups: the group generated by the face pairings of the fundamental polyhedron over a closed chain of $2n$ circles in which each circle is externally tangent to its two neighbors, so that the $2n$ tangency points become parabolic fixed points (fixed points of Möbius transformations conjugate to translations). The common boundary $L$ of the two invariant domains of such a group is its limit set — a Jordan curve through the tangency points, which Poincaré called the limit line and described in Cantor's set-theoretic terminology as the union of the point set $P$ of parabolic fixed points and its first derived set (set of accumulation points). He proved that $L$ has a tangent at every point of $P$, that chords joining two points of $P$ need not converge to that tangent, and that $L$ has no osculating circle (no circle of second-order contact) at those parabolic points, and he conjectured: 'I have every reason to believe that there is no tangent at the points of $L$ that are not part of $P$.' In modern terms these groups are the necklace groups — free, geometrically finite Kleinian groups with two invariant components, whose limit set is a quasicircle — and by the Beardon–Maskit limit-point classification the points of $L\setminus P$ are exactly the conical limit points, so the conjecture asserts that the limit quasicircle of a non-Fuchsian necklace group is nowhere tangent except at its cusps. No proof, disproof, or modern restatement of this pointwise conjecture is known, and it remains open.

Background

A Kleinian group is a discrete group of Möbius transformations — fractional linear transformations $z\mapsto (az+b)/(cz+d)$ of the Riemann sphere $\hat{\mathbb{C}} = \mathbb{C}\cup\{\infty\}$ — in matrix terms the action of $\mathrm{PSL}(2,\mathbb{C})$, the complex $2\times 2$ matrices of determinant $1$ up to sign — acting properly discontinuously on some nonempty open subset of the sphere. The complement of the largest such open set is the limit set $L$ of the group: the set of accumulation points of any orbit, and the locus where discontinuity fails. If all elements of the group map a common circle to itself, the group is called Fuchsian (after Lazarus Fuchs) and its limit set lies on that circle; the Kleinian case, named after Felix Klein, is the general one. In an 1883 memoir in Acta Mathematica, Henri Poincaré undertook the first systematic construction and classification of Kleinian groups beyond the Fuchsian case. His method is geometric. A configuration of circles in the plane is lifted to hemispheres orthogonal to the sphere at infinity in hyperbolic 3-space (the unique simply connected 3-dimensional space of constant negative curvature, realized as the upper half-space with metric $ds^2 = (dx_1^2+dx_2^2+dx_3^2)/x_3^2$); the hemispheres bound a fundamental polyhedron — a region whose images under the generated group tile the space — and discontinuity is verified by checking the edge and vertex cycles of the polyhedron. Friedrich Schottky had earlier constructed the first examples, groups generated by transformations pairing disjoint circles (Schottky groups), and Poincaré's historical section credits him with the first observation of discontinuity for symmetric groups of this kind, as well as Klein's observation of discontinuity domains bounded by infinitely many circles. The polyhedron method was made into a general rigorous tool in Bernard Maskit's Poincaré polyhedron theorem, and the groups admitting finite-sided fundamental polyhedra — the class to which all of Poincaré's examples belong — became the geometrically finite groups, whose structure theory begins with Lars Ahlfors's finiteness results.

The memoir organizes its examples by species (first species: deformable into a Fuchsian group; second species: not), by the genus of the generating polygon, and into families counted by the number of second-kind faces — faces of the polyhedron lying over regions of the plane complementary to the circle configuration, which are not paired. The second family (§8) is built from $2n$ circles $C_1,\dots,C_{2n}$ whose closed disks are pairwise disjoint except that each $C_i$ is externally tangent to its two cyclic neighbors, at $2n$ distinct points $A_i$; the interstice $R$ inside the chain and the exterior region $R'$ form the two second-kind faces, and the $A_i$ are isolated vertices of the polyhedron organized into parabolic cycles (a vertex cycle is parabolic when the product of the pairing transformations around the vertex is a parabolic Möbius transformation, one conjugate to a translation $z\mapsto z+1$). Poincaré proves that the plane decomposes into two invariant domains $D\supset R$ and $D'\supset R'$, 'separated by a line $L$, if one can call that a line'. This $L$ is the limit set, a compact, perfect (closed with no isolated points), connected subset of the sphere — in fact a Jordan curve. Poincaré describes it with the then-new set-theoretic vocabulary of Georg Cantor (cited in a footnote): the translates of the $2n$ tangency points form an infinite point set $P$ (Cantor's 'Punktmenge'), and $L$ is obtained by adding Cantor's first derived set $P'$ (the set of accumulation points): $L = P\cup P'$, 'a perfect and connected point set'. The points of $P$ are exactly the parabolic fixed points of the group. Modern authors call these groups and their limit sets necklaces, and the kissing-Schottky description is the same class: a free Schottky-type group whose pairing circles have been brought into tangency. The limit curve is a quasicircle — the image of a round circle under a quasiconformal homeomorphism of the sphere, that is, a homeomorphism distorting infinitesimal circles to ellipses of uniformly bounded eccentricity. The historian Gregory Moore has traced how the set-theoretic concepts Poincaré invoked here were then entering analysis generally.

For a compact plane set and a point $A$ of it, Poincaré says the set has a tangent at $A$ if there is a line through $A$ that is the limit of the secants (chords) $AB$ joining $A$ to a variable point $B$ of the set as $B\to A$; in polar coordinates $(\rho,\omega)$ with pole $A$ and polar axis that line, this means $\omega\to 0$ whenever $\rho\to 0$. For a Jordan curve this is equivalent to the two one-sided tangent directions at $A$ existing and lying on one line; it is deliberately weaker than requiring chords $BC$ with two variable endpoints $B,C$ of the set, both tending to $A$, to converge. In §8 Poincaré proves three things about the second family's limit line. First, every point of $P$ is a tangent point: at a tangency point $A_0$ whose parabolic cycle is a single vertex, with $S$ the parabolic generator fixing $A_0$ (conjugate to $z\mapsto z+h$) and $N$ the point with $\arg(N-A_0) = -\arg h$, the line $A_0N$ is the tangent. The proof constructs two circles $K$ and $K'$ externally tangent at $A_0$ and both tangent to the line $A_0N$, and shows that the iterates of $S$ swallow the interiors of $K$ and $K'$ into the two invariant domains, so that near $A_0$ the limit line is confined to the two cusp-shaped regions exterior to both circles — the ancestor of the modern horoball argument at a parabolic point (a horoball at a boundary point being the limit of hyperbolic balls internally tangent to the sphere at infinity at that point). Second, the tangent at $A_0$ is nonetheless not a limit of chords with two moving endpoints: if $B,C\in P$ both tend to $A_0$, the lines $BC$ need not converge to $A_0N$ (Poincaré iterates a circle through three points of $P$ under powers of $S$ and computes the limiting angle). Third, $L$ has no osculating circle at $A_0$ — no circle of second-order contact: the circles tangent to $A_0N$ at $A_0$ and passing through a variable point $B\in L$ have no limit as $B\to A_0$, by the same iteration. He then continues: 'De plus j'ai tout lieu de croire qu'il n'y a pas de tangente aux points de $L$ qui ne font pas partie de $P$' — 'Moreover, I have every reason to believe that there is no tangent at the points of $L$ that are not part of $P$' — and closes the discussion: 'I have said enough, I think, to make understood to what point the line $L$ differs from an analytic curve.'

The modern theory identifies the construction exactly. The second-family groups are the necklace groups: free, geometrically finite Kleinian groups with two simply connected invariant components, whose parabolic subgroups are cyclic (rank one), and whose limit set is a Jordan curve through the parabolic points. Robert Riley began the modern computational study of discrete groups generated by parabolic transformations, and the book-length computational treatment by David Mumford, Caroline Series, and David Wright displays these groups' limit sets as the eponymous necklaces. The structural fact that fixes the modern reading of the conjecture is the limit-point classification of Alan Beardon and Bernard Maskit: for a geometrically finite group, every limit point is either conical — some orbit approaches it while staying within bounded hyperbolic distance of the geodesic ray aimed at it — or bounded parabolic — a parabolic fixed point whose parabolic stabilizer acts cocompactly on the limit set with that point removed. For a necklace group, the points of Poincaré's set $P$ are the bounded parabolic points — his two circles $K$ and $K'$ are exactly the boundary of a horoball neighborhood on which the parabolic generator acts cocompactly (with compact quotient) — and $L\setminus P$ is the set of conical limit points. The conjecture therefore reads: every conical limit point of a necklace group fails to be a tangent point of the limit quasicircle. One degenerate subcase must be excluded: if the chain is symmetric under inversion in a common circle (all $C_i$ orthogonal to one circle $\Gamma$), then the group is conjugate to a Fuchsian group and $L = \Gamma$ is a round circle, on which every point is a tangent point; the conjecture is a statement about the non-Fuchsian configurations.

Several deep theories bear on the local geometry of such curves without answering Poincaré's question. Nikolai Makarov proved that harmonic measure — the hitting distribution on the boundary of Brownian motion started inside a domain — on the boundary of any simply connected plane domain is supported on a set of Hausdorff dimension exactly one (Hausdorff dimension being the notion of size defined by covering the set with sets of vanishing diameter); this universal theorem applies to the necklace group's domains, but it is a statement about a measure, not about every point. Joan Josep Carmona and Christian Pommerenke developed the quantitative theory of twisting of conformal maps: at a twist point of a boundary curve the curve turns through unbounded angle in every neighborhood, the opposite of having a tangent, and their results quantify this behavior almost everywhere (for harmonic measure) and in the sense of dimension; Fausto Di Biase, Bert Fischer, and Rüdiger Urbanke analyzed the twist points of the von Koch snowflake, a self-similar quasicircle, from this point of view. These almost-everywhere and dimension statements do not give the pointwise dichotomy Poincaré conjectured. Christopher Bishop and Peter Jones proved sharp results on the Hausdorff dimension of limit sets of finitely generated Kleinian groups, including criteria for the dimension-one case — metric information that likewise does not see the direction of a tangent. The memoir's own classification program has been transformed beyond recognition: its reflection-symmetric cases belong to the modern theory of discrete hyperbolic reflection groups surveyed by E. B. Vinberg, and the classification of finitely generated Kleinian surface groups (Kleinian groups isomorphic to the fundamental group of a closed surface) was completed in a different sense by the ending lamination theorem of Jeffrey Brock, Richard Canary, and Yair Minsky. None of this modern theory addresses the pointwise tangent question, which survives untouched.

Problem Statement

The problem is Poincaré's tangent conjecture for the second family of Kleinian groups (1883 memoir, §8).

Setup. Let $n\geq 2$ and let $C_1,\dots,C_{2n}$ be circles in the Riemann sphere forming a closed chain: the closed disks they bound are pairwise disjoint except that $C_i$ is externally tangent to $C_{i+1}$ (indices modulo $2n$) at a point $A_i$, and these $2n$ tangency points are distinct. Construct the group as in §8: lift the circles to hemispheres orthogonal to the sphere at infinity in hyperbolic 3-space, take the fundamental polyhedron whose first-kind faces lie over the $C_i$ and whose two second-kind faces lie over the interstice $R$ inside the chain and the exterior region $R'$, and let $G$ be the group generated by the face-pairing Möbius transformations, the vertex cycles over the $A_i$ being parabolic. In modern terms $G$ is a necklace group: a free, geometrically finite Kleinian group with two simply connected invariant domains $D\supset R$ and $D'\supset R'$, all of whose parabolic fixed points lie on the common boundary $L = \partial D = \partial D'$, a Jordan curve (indeed a quasicircle). Let $P\subset L$ denote the set of parabolic fixed points of $G$ — the $G$-translates of the tangency points $A_i$ — a countable dense subset of $L$; Poincaré writes $L = P\cup P'$ with $P'$ the first derived set (set of accumulation points).

Tangent, Poincaré's definition. A point $A\in L$ is a tangent point of $L$ if there is a line $\ell$ through $A$ such that the secant lines $AB$, $B\in L\setminus\{A\}$, converge to $\ell$ as $B\to A$; in polar coordinates $(\rho,\omega)$ with pole $A$ and axis $\ell$: $\omega\to 0$ whenever $\rho\to 0$.

What is already proved. Poincaré proves that every point of $P$ is a tangent point of $L$; that at a point $A\in P$ the chords $BC$ with $B,C\in P$, $B,C\to A$, need not converge to the tangent; and that $L$ has no osculating circle at points of $P$ (the circles tangent to the tangent line at $A$ through a variable point $B\in L$ have no limit as $B\to A$). These parts of §8 are not at issue.

The open question. Poincaré's conjecture, verbatim: 'De plus j'ai tout lieu de croire qu'il n'y a pas de tangente aux points de $L$ qui ne font pas partie de $P$' ('Moreover, I have every reason to believe that there is no tangent at the points of $L$ that are not part of $P$'). By the Beardon–Maskit classification the points of $L\setminus P$ are exactly the conical limit points of $G$, so the conjecture is equivalent to: no conical limit point of a necklace group is a tangent point of the limit quasicircle. Prove, or disprove: for every $n\geq 2$ and every non-Fuchsian second-family configuration as above, the set of tangent points of $L$ is exactly $P$.

Required precision. The conjecture is false as stated for the Fuchsian degeneration of the construction: if all $2n$ circles of the chain are orthogonal to one common circle $\Gamma$ — equivalently, if $G$ is conjugate to a Fuchsian group, so that $L$ is the round circle $\Gamma$ — then every point of $L\setminus P$ is a tangent point. The conjecture must therefore be understood for the non-Fuchsian configurations (equivalently, whenever $L$ is not a round circle).

References

1. H. Poincaré, 'Mémoire: Les groupes kleinéens', Acta Mathematica 3 (1883) 49–92. DOI: 10.1007/BF02422441, https://doi.org/10.1007/BF02422441
2. F. Schottky, 'Ueber die conforme Abbildung mehrfach zusammenhängender ebener Flächen', Journal für die reine und angewandte Mathematik 83 (1877) 300–351. DOI: 10.1515/crll.1877.83.300, https://doi.org/10.1515/crll.1877.83.300
3. B. Maskit, 'On Poincaré's theorem for fundamental polygons', Advances in Mathematics 7 (1971) 219–230. DOI: 10.1016/S0001-8708(71)80003-8, https://doi.org/10.1016/S0001-8708(71)80003-8
4. L. V. Ahlfors, 'Fundamental polyhedrons and limit point sets of Kleinian groups', Proceedings of the National Academy of Sciences 55 (1966) 251–254. DOI: 10.1073/PNAS.55.2.251, https://doi.org/10.1073/PNAS.55.2.251
5. R. Riley, 'A quadratic parabolic group', Mathematical Proceedings of the Cambridge Philosophical Society 77 (1975) 281–288. DOI: 10.1017/S0305004100051094, https://doi.org/10.1017/S0305004100051094
6. A. F. Beardon and B. Maskit, 'Limit points of Kleinian groups and finite sided fundamental polyhedra', Acta Mathematica 132 (1974) 1–12. DOI: 10.1007/BF02392106, https://doi.org/10.1007/BF02392106
7. E. B. Vinberg, 'Hyperbolic reflection groups', Russian Mathematical Surveys 40 (1985) 31–75. DOI: 10.1070/RM1985V040N01ABEH003527, https://doi.org/10.1070/RM1985V040N01ABEH003527
8. N. G. Makarov, 'On the Distortion of Boundary Sets Under Conformal Mappings', Proceedings of the London Mathematical Society (3) 51 (1985) 369–384. DOI: 10.1112/plms/s3-51.2.369, https://doi.org/10.1112/plms/s3-51.2.369
9. J. J. Carmona and C. Pommerenke, 'Twisting Behaviour of Conformal Maps', Journal of the London Mathematical Society (2) 56 (1997) 16–36. DOI: 10.1112/S0024610797005334, https://doi.org/10.1112/S0024610797005334
10. F. Di Biase, B. Fischer, and R. Urbanke, 'Twist points of the von Koch snowflake', Proceedings of the American Mathematical Society 126 (1998) 1487–1490. DOI: 10.1090/S0002-9939-98-04226-9, https://doi.org/10.1090/S0002-9939-98-04226-9
11. C. J. Bishop and P. W. Jones, 'Hausdorff dimension and Kleinian groups', Acta Mathematica 179 (1997) 1–39. DOI: 10.1007/BF02392718, https://doi.org/10.1007/BF02392718
12. D. Mumford, C. Series, and D. Wright, 'Indra's Pearls: The Vision of Felix Klein', Cambridge University Press, 2002. DOI: 10.1017/cbo9781107050051, https://doi.org/10.1017/cbo9781107050051
13. G. H. Moore, 'The emergence of open sets, closed sets, and limit points in analysis and topology', Historia Mathematica 35 (2008) 220–241. DOI: 10.1016/j.hm.2008.01.001, https://doi.org/10.1016/j.hm.2008.01.001
14. J. Brock, R. Canary, and Y. Minsky, 'The classification of Kleinian surface groups, II: The Ending Lamination Conjecture', Annals of Mathematics 176 (2012) 1–149. DOI: 10.4007/annals.2012.176.1.1, https://doi.org/10.4007/annals.2012.176.1.1