Structure and conjugacy classification of infinitely generated Fuchsian groups — the class Poincaré set aside in 1882

Poincaré's founding memoir « Théorie des groupes fuchsiens » (Acta Mathematica 1, 1882) restricts itself to Fuchsian groups derivable from finitely many fundamental substitutions, declaring that groups for which this is impossible are 'left systematically aside'. Four of the issues the memoir leaves open — the theory of Kleinian groups, the exposition of the Fuchsian (automorphic) functions that integrate linear differential equations with algebraic coefficients, the explicit fundamental-polygon constructions for every family and order, and the concrete solution of the coefficient inequalities — were closed by Poincaré's own subsequent memoirs (1882–1908) and by the classical theory (Fricke–Klein as recorded by Macbeath; Keen; Maskit), which classifies finitely generated Fuchsian groups by signature plus Teichmüller parameters. The surviving open core is the first exclusion: a structure theory and conjugacy classification, up to conjugacy in $\mathrm{PSL}(2,\mathbb{R})$ (the orientation-preserving isometry group of the hyperbolic plane), of Fuchsian groups that are not finitely generated. Existence is easy and the quotient surfaces are classified topologically (Kerékjártó–Richards), but no conjugacy invariants or normal-form theory exist; recent descriptive-set-theoretic work shows the conjugacy relation on all torsion-free Fuchsian groups is of maximal countable-structure complexity (Bergfalk–Smythe) while the finitely generated subcase is concretely classifiable. The open problem is to determine which subclasses admit usable classification — exact Borel complexity of natural subclasses, complete conjugacy invariants, canonical infinitely-sided fundamental domains, and the moduli theory of infinite-type hyperbolic surfaces — jointly constituting a structure theory for the excluded class.

Background

Setting. A Fuchsian group is a discrete subgroup of $\mathrm{PSL}(2,\mathbb{R})$, the group of orientation-preserving isometries of the hyperbolic plane $\mathbb{H}^2$ (the unique simply connected complete surface of constant curvature $-1$, modeled for instance by the upper half-plane with metric $|dz|^2/(\operatorname{Im} z)^2$). The quotient $\mathbb{H}^2/\Gamma$ is a hyperbolic surface, with cone points if $\Gamma$ has torsion. The classical way to describe such a group is by a fundamental polygon: a convex region $R_0\subset\mathbb{H}^2$ whose $\Gamma$-translates tile the plane, with Möbius transformations pairing the sides of $R_0$ (Poincaré's « substitutions fondamentales »); the side-pairing transformations generate $\Gamma$.\n\nThe source. In the founding memoir « Théorie des groupes fuchsiens » (Acta Mathematica 1 (1882) 1–62), Poincaré organized this theory: he classified possible generating polygons into families and orders, gave existence conditions for the corresponding groups (his polygon theorem), and reduced the determination of the coefficients of the fundamental substitutions to explicit algebraic inequalities (with $ad-bc=1$ imposed), worked out only for representative examples such as the octagon and the hexagon. Two restrictions are declared openly. First (Œuvres, t. II, p. 117): « On peut concevoir aussi des groupes tels qu'il soit impossible de les faire dériver d'un nombre fini de substitutions fondamentales. Mais nous les laisserons systématiquement de côté » — groups not derivable from finitely many fundamental substitutions are systematically set aside. Second (Œuvres, t. II, p. 166): the « groupes kleinéens » — discrete subgroups of $\mathrm{PSL}(2,\mathbb{C})$ preserving no common circle — are reserved for a special memoir; and the whole function-theoretic side (the automorphic, 'Fuchsian', functions integrating linear differential equations with algebraic coefficients) is postponed to later memoirs.\n\nWhat is classical for the finitely generated case. A finitely generated Fuchsian group is classified up to isomorphism by its signature $(g;m_1,\dots,m_r)$ — the genus and cone-point orders of the quotient — a classification essentially solved in the classical Fricke–Klein theory, as recorded in Macbeath's 1967 introduction, which also extends it to groups containing reflections. Poincaré's coefficient inequalities become the statement that realizations of a fixed signature form the Teichmüller space (the space of marked conformal structures on the surface — a finite-dimensional cell in this case), so the finitely generated world is completely classified: discrete combinatorial data plus finitely many real parameters. Keen (1966) produced canonical fundamental polygons, and Maskit (1971) gave the first complete proof of Poincaré's polygon theorem. In dimension 2 the finiteness conditions collapse: a Fuchsian group is finitely generated if and only if it is geometrically finite (admits a finite-sided fundamental polygon) if and only if its quotient is topologically finite.\n\nThe excluded class. A Fuchsian group requiring infinitely many generators has a quotient surface of infinite topological type (infinite genus or infinitely many ends). Such groups exist in abundance — for example, the subgroup of $\mathrm{PSL}(2,\mathbb{R})$ uniformizing the infinite cyclic cover of a closed hyperbolic surface, i.e. the kernel of an epimorphism from the closed surface's fundamental group onto $\mathbb{Z}$, which is a free group of countably infinite rank. The quotient surfaces are classified topologically by the Kerékjártó–Richards theorem: a noncompact surface is determined by its genus, its number of boundary components, and the pair consisting of its space of ends (an end is an equivalence class of exiting nested neighborhoods; the set of ends is a closed subset of a Cantor set) together with the closed subspace of ends accumulated by genus. This classifies only the topology, not the group up to conjugacy in $\mathrm{PSL}(2,\mathbb{R})$: within each topological type there remains an infinite-dimensional family of conformal structures, and for infinite-type surfaces even the basic Teichmüller/moduli theory has only recently begun to be constructed. The logically precise modern form of the classification question is descriptive-set-theoretic: endow the class of (say, torsion-free) Fuchsian groups with a standard Borel structure (e.g. via the Chabauty/Effros–Borel space of closed subgroups) and ask for the complexity of the conjugacy relation up to Borel reducibility (a Borel map between standard Borel spaces carrying equivalent points to equivalent points and non-equivalent points to non-equivalent points). A relation 'concretely classifiable' (Borel bireducible with equality of reals $=_{\mathbb{R}}$) admits real-number complete invariants; 'essentially countably universal' means maximally complex among equivalence relations classifiable by countable structures. Bergfalk–Smythe (2025) proved that conjugacy on all discrete torsion-free subgroups of any noncompact semisimple Lie group — hence on all torsion-free Fuchsian groups — is essentially countably universal, while for finitely generated Fuchsian groups it is exactly $=_{\mathbb{R}}$ (Peterzil (2026) extended this to groups with torsion). Consequently a signature-like discrete classification of the entire excluded class is impossible, and the open problem is to determine how much of that class — and which of its subclasses — can be classified and structurally organized.

Problem Statement

The task is to do for the class Poincaré 'left systematically aside' in 1882 what his memoir and its successors did for the finitely generated class: build the structure theory of Fuchsian groups that are not finitely generated, up to conjugacy in $\mathrm{PSL}(2,\mathbb{R})$. The overarching objective is a classification-and-normal-form theory for this class, with three required components.\n\n(a) Conjugacy classification and its limits. Determine which subclasses of the infinitely generated Fuchsian groups admit usable conjugacy invariants. In particular, compute the exact Borel complexity of the conjugacy relation on the class of all infinitely generated Fuchsian groups and on at least the following subclasses — torsion-free groups; groups of the first kind (whose limit set, the set of accumulation points of an orbit on the circle at infinity, is the entire circle) versus the second kind (proper limit set); groups whose quotient surface has a fixed homeomorphism type (the last to be treated uniformly in the type, i.e. for every homeomorphism type of an infinite-type surface, not a single chosen type) — up to Borel bireducibility (two equivalence relations are Borel bireducible when Borel maps reduce each to the other, carrying equivalent points to equivalent points and conversely), and exhibit complete effective invariants for each of these subclasses where they exist. The known extremes are the finitely generated case, which is concretely classifiable (Borel bireducible with equality of real numbers, $=_{\mathbb{R}}$; Bergfalk–Smythe Theorem 7.2, extended to torsion by Peterzil 2026), and the class of all torsion-free Fuchsian groups, which is essentially countably universal (maximally complex among equivalence relations classifiable by countable structures; Bergfalk–Smythe Corollary 6.5). The complexity of the infinitely generated class and of its subclasses — the interior between these extremes — is undetermined.\n\n(b) Normal forms. Construct canonical fundamental domains for these groups — distinguished infinitely-sided polygons, or controlled exhaustions by finite-sided domains with explicit side-pairing data — paralleling the canonical-polygon theory of the finitely generated case (Keen 1966) and the polygon theorem (Maskit 1971), but valid for all groups of one of the subclasses named in (a) rather than isolated model surfaces (the Loch Ness monster, the Cantor tree, and the blooming Cantor tree — the three standard model surfaces of infinite type: infinite genus with a single end; planar with a Cantor set of ends; and a Cantor set of ends every point of which is accumulated by genus — as uniformized by Arredondo–Ramírez Maluendas 2020).\n\n(c) The associated moduli theory. Develop the Teichmüller/moduli theory of infinite-type hyperbolic surfaces — the geometric form of the conjugacy classification — beyond the first steps (the marked moduli space of Nielsen-convex structures of Tappu 2023, where a complete hyperbolic surface is Nielsen-convex when its convex core fills the whole surface, equivalently when its universal-cover group is of the first kind; the generic spectral-rigidity results of Baik–Choi–Kim 2020), including the structure of the resulting non-locally-compact, generally non-Hausdorff moduli spaces.\n\nComponents (a), (b), and (c) are three faces of one objective — (c) is the geometric reformulation of (a), and (b) is its group-theoretic normal-form aspect — and a complete resolution requires all three. A single signature-like discrete classification of the entire class is now known to be impossible (essential countable universality of conjugacy on the torsion-free class, Bergfalk–Smythe 2025), so the completion boundary of the problem is precisely, for the class of all infinitely generated Fuchsian groups and each of the natural subclasses named in (a): the maximal classifiable subclasses, the invariants that organize the remainder, and the exact complexity of the whole class — not a proxy restricted to finitely many model surfaces.

References

1. H. Poincaré, « Théorie des groupes fuchsiens », Acta Mathematica 1 (1882), 1–62. DOI: 10.1007/BF02592124. https://doi.org/10.1007/BF02592124
2. H. Poincaré, Œuvres de Henri Poincaré, tome II : Fonctions fuchsiennes (Gauthier-Villars, Paris, 1916); « Théorie des groupes fuchsiens » at pp. 108–168. Digitized scan (Internet Archive): https://archive.org/details/vresdehenripoin02poin; PDF served by the Archives Henri Poincaré: https://henripoincare.fr/files/original/03d1277ebc8ed5a59bb1650cef698fad15399710.pdf
3. H. Poincaré, « Mémoire sur les fonctions fuchsiennes », Acta Mathematica 1 (1882), 193–294. DOI: 10.1007/BF02592135. https://doi.org/10.1007/BF02592135
4. H. Poincaré, « Mémoire : Les groupes kleinéens », Acta Mathematica 3 (1883), 49–92. DOI: 10.1007/BF02422441. https://doi.org/10.1007/BF02422441
5. H. Poincaré, « Sur les groupes des équations linéaires », Acta Mathematica 4 (1884), 201–312. DOI: 10.1007/BF02418420. https://doi.org/10.1007/BF02418420
6. H. Poincaré, « Mémoire sur les fonctions zétafuchsiennes », Acta Mathematica 5 (1884), 209–278. DOI: 10.1007/BF02421560. https://doi.org/10.1007/BF02421560
7. H. Poincaré, « Sur l'uniformisation des fonctions analytiques », Acta Mathematica 31 (1908), 1–63. DOI: 10.1007/BF02415442. https://doi.org/10.1007/BF02415442
8. Lipman Bers, « Automorphic forms and Poincaré series for infinitely generated Fuchsian groups », American Journal of Mathematics 87 (1965), no. 1, 196–214. DOI: 10.2307/2373231. https://doi.org/10.2307/2373231
9. Ian Richards, « On the classification of noncompact surfaces », Transactions of the American Mathematical Society 106 (1963), 259–269. DOI: 10.1090/S0002-9947-1963-0143186-0. https://doi.org/10.1090/S0002-9947-1963-0143186-0
10. A. M. Macbeath, « The classification of non-euclidean plane crystallographic groups », Canadian Journal of Mathematics 19 (1967), 1192–1205. DOI: 10.4153/CJM-1967-108-5. https://doi.org/10.4153/CJM-1967-108-5
11. Linda Keen, « Canonical polygons for finitely generated Fuchsian groups », Acta Mathematica 115 (1966), 1–16. DOI: 10.1007/BF02392200. https://doi.org/10.1007/BF02392200
12. Bernard 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
13. William P. Thurston, « Three dimensional manifolds, Kleinian groups and hyperbolic geometry », Bulletin of the American Mathematical Society 6 (1982), 357–382. DOI: 10.1090/S0273-0979-1982-15003-0. https://doi.org/10.1090/S0273-0979-1982-15003-0
14. Pekka Tukia, « On torsionless subgroups of infinitely generated Fuchsian groups », Annales Academiae Scientiarum Fennicae, Series A I Mathematica 4 (1979), 203–205. DOI: 10.5186/AASFM.1978-79.0428. https://doi.org/10.5186/AASFM.1978-79.0428
15. Shunsuke Morosawa, « Limit points for infinitely generated Fuchsian groups », Mathematical Proceedings of the Cambridge Philosophical Society 104 (1988), 539–545. DOI: 10.1017/S0305004100065737. https://doi.org/10.1017/S0305004100065737
16. Javier Aramayona and Nicholas G. Vlamis, « Big mapping class groups: an overview », in: In the Tradition of Thurston: Geometry and Topology (Springer, 2020), chapter 12, pp. 459–496. DOI: 10.1007/978-3-030-55928-1_12; arXiv:2003.07950. https://arxiv.org/abs/2003.07950
17. John A. Arredondo and Camilo Ramírez Maluendas, « On infinitely generated Fuchsian groups of the Loch Ness monster, the Cantor tree and the blooming Cantor tree », Complex Manifolds 7 (2020), 73–92. DOI: 10.1515/coma-2020-0004; arXiv precursor « On infinitely generated Fuchsian groups of some infinite genus surfaces », arXiv:1806.04492. https://arxiv.org/abs/1806.04492
18. Hyungryul Baik, Inhyeok Choi, and Dongryul M. Kim, « Simple length spectra as moduli for hyperbolic surfaces and rigidity of length identities », arXiv:2012.05652 (2020), to appear in Annales de l'Institut Fourier. https://arxiv.org/abs/2012.05652
19. Chaitanya Tappu, « A moduli space of marked hyperbolic structures for big surfaces », arXiv:2311.01551 (2023). https://arxiv.org/abs/2311.01551
20. Jeffrey Bergfalk and Iian B. Smythe, « Manifold classification from the descriptive viewpoint », arXiv:2512.24996 (2025, preliminary version). https://arxiv.org/abs/2512.24996
21. George Peterzil, « Classification of Fuchsian groups with torsion », arXiv:2606.31459 (2026). https://arxiv.org/abs/2606.31459