Manifold structure of the connected components of Weil's space $\Re_0(\Gamma,G)$ of faithful discrete cocompact representations

For a finitely generated discrete group $\Gamma$ and a connected Lie group $G$ (a group that is simultaneously a smooth manifold with smooth group operations), André Weil (1960) studied the representation space $\Re(\Gamma,G)$ of all group homomorphisms $\Gamma\to G$ endowed with the topology of pointwise convergence, and the subset $\Re_0(\Gamma,G)$ of injective representations whose image is discrete with compact quotient $G/r(\Gamma)$. He proved that $\Re_0(\Gamma,G)$ is open in $\Re(\Gamma,G)$ and that two representations lying in a common connected component of $\Re_0(\Gamma,G)$ yield analytically homeomorphic quotient manifolds — but left open the local geometric structure of $\Re_0$: whether each connected component of $\Re_0(\Gamma,G)$ is itself a manifold, the classical positive example being the Fuchsian case ($\Gamma$ a cocompact surface group, $G=PSL(2,\mathbb{R})$, the orientation-preserving isometry group of the hyperbolic plane), where the components are Teichmüller spaces (the classical deformation spaces of marked conformal structures on surfaces). Subsequent literature resolves major classes — Fuchsian groups; lattices in simply connected solvable Lie groups (Wang 1963); locally rigid lattices, where components collapse to single conjugation orbits — yet no resolution was found for general solvable $G$ or for mixed-type $G$ carrying both a nontrivial semisimple Levi factor and a nontrivial solvable radical (the radical being the largest connected solvable normal subgroup of $G$, a Levi factor a connected semisimple subgroup complementary to it). This problem asks for a determination of the pairs $(\Gamma,G)$ for which the connected components of $\Re_0(\Gamma,G)$ are real-analytic manifolds, equivalently for which the representation space $\Re(\Gamma,G)$ is a manifold germ at every faithful discrete cocompact representation.

Background

Definitions. A Lie group is a group that is simultaneously a smooth (here: real-analytic) manifold such that multiplication and inversion are smooth. A discrete subgroup $\Gamma\subset G$ is a lattice if the quotient space $G/\Gamma$ has finite invariant volume, and a uniform (or cocompact) lattice if $G/\Gamma$ is compact. For a finitely generated group $\Gamma$ (one generated by a finite set $S$) and a Lie group $G$, the representation space $\Re(\Gamma,G)=\mathrm{Hom}(\Gamma,G)$ is the set of all group homomorphisms $\Gamma\to G$ with the topology of pointwise convergence; choosing the finite generating set $S$ identifies $\Re(\Gamma,G)$ with the subset of the finite product $G^S$ cut out by the finitely many relator equations of a finite presentation of $\Gamma$, so $\Re(\Gamma,G)$ is a real-analytic subset of $G^S$ (a locus defined near each point by finitely many real-analytic equations), a structure independent of $S$ up to analytic isomorphism [1][9]. Write $\Re_0(\Gamma,G)\subset\Re(\Gamma,G)$ for the locus of representations $r$ that are injective (faithful), whose image $r(\Gamma)$ is a discrete subgroup of $G$, and for which the right-coset space $G/r(\Gamma)$ is compact — the faithful uniform-lattice embeddings. A connected component is a maximal connected subset; a real-analytic manifold is a space locally modeled on open subsets of some fixed Euclidean space by real-analytic coordinate changes.

Weil's results (1960) [1]. First, $\Re_0(\Gamma,G)$ is an open subset of $\Re(\Gamma,G)$ (as summarized, e.g., in [8]: 'for any cocompact lattice $\Delta$ in a Lie group $H$ the space $X(\Delta,H)$ [of lattice embeddings] is an open and locally path connected subset of $\mathrm{Hom}(\Delta,H)$'). Second, if $r,r'\in\Re_0(\Gamma,G)$ lie in the same connected component, then there is an analytic homeomorphism $F:G\to G$ with $F(g\cdot r(\gamma))=F(g)\cdot r'(\gamma)$ for all $g\in G,\gamma\in\Gamma$, descending to an analytic homeomorphism $G/r(\Gamma)\to G/r'(\Gamma)$ of the compact quotient manifolds. These results make the connected components of $\Re_0$ the natural parameter spaces for deformations of a compact quotient; the paper nevertheless leaves open the local analytic geometry of these components. Weil's companion paper (1962) [2] proves a different, cohomological rigidity statement: $H^1(\Gamma,\mathfrak{g})=0$ (first group cohomology with coefficients in the Lie algebra $\mathfrak{g}$ of $G$, twisted by the adjoint action) for uniform irreducible lattices in semisimple $G$ not locally isomorphic to $SL(2,\mathbb{R})$, which yields local rigidity — the inclusion $\Gamma\hookrightarrow G$ is isolated up to conjugation. Part I (1960) and Part II (1962) are distinct and must not be conflated.

Known cases of the manifold question. (a) Fuchsian case: for a closed orientable surface $\Sigma_g$ of genus $g\ge 2$ with $\Gamma=\pi_1\Sigma_g$ and $G=PSL(2,\mathbb{R})$ (the orientation-preserving isometry group of the hyperbolic plane), the discrete-faithful locus is the union of two connected components of $\mathrm{Hom}(\Gamma,G)$, and these components and their quotient — the Teichmüller space $T(\Sigma_g)$, the manifold of marked conformal structures on $\Sigma_g$ — are connected manifolds describable by algebraic equalities and inequalities (a description going back to Fricke and Klein; Goldman's Euler-class classification of components [5]; Komori's semialgebraic description [13]; summary in [8]). (b) Solvable case: a lattice in a simply connected solvable Lie group (one whose commutator series terminates) has deformation components that are manifolds — Wang [3] identified the component of the inclusion $X(\Gamma,G)^0$ with the identity component of a suitable Lie-group automorphism group via restriction of automorphisms, as stated and used in [8]. For solvable $G$ that is not simply connected, $X(\Gamma,G)$ may have infinitely many connected components [8], and manifold-ness of the components is not settled in the literature. (c) Locally rigid case: if the inclusion is locally rigid (e.g., $G$ semisimple without $SL(2,\mathbb{R})$-type factors, by [2] with Selberg and Calabi; non-uniform rank-one lattices by Garland–Raghunathan [18]; higher rank via Margulis superrigidity [19] — see the historical accounts in [10][11]), then near a locally rigid point the component of $\Re_0$ is a single conjugation orbit $G/Z_G(r(\Gamma))$ ($Z_G$ the centralizer), which is a manifold.

General framework for the obstruction. Because $\Re_0(\Gamma,G)$ is open in the real-analytic set $\Re(\Gamma,G)$, a connected component of $\Re_0(\Gamma,G)$ is a manifold exactly when $\Re(\Gamma,G)$ is a manifold germ at every point of $\Re_0(\Gamma,G)$. The study of singularities and smooth points of representation varieties is classical: Lubotzky–Magid [6] develop the general theory (dimension and smoothness criteria at a representation in terms of the cohomology of $\Gamma$, with formal tangent space given by the cocycle space $Z^1(\Gamma,\mathfrak{g}_{\mathrm{Ad}\, r})$ and obstructions governed by $H^2$), and Goldman's survey [9] records that $\mathrm{Hom}(\pi_1\Sigma,G)$ 'may have singularities, and not be a manifold', is stratified into smooth submanifolds, and that its conjugation quotient may have orbifold singularities or fail to be Hausdorff — so Weil's question is a genuine restriction of a generally singular object to the uniform-lattice locus. Notably, some later authors write loosely that the 'deformation space $\Re(\Gamma,G)$ has the structure of a manifold' when only local arcwise-connectedness of the real-analytic set is used (e.g., in the proof of Proposition 2.5 in [10]). Adjacent modern activity includes: topological local rigidity of uniform lattices in compactly generated groups [11]; discreteness of deformations of cocompact discrete subgroups [12]; the smoothness question for the deformation space of complete affine structures on the 2-torus, where period coordinates exhibit it as $\mathbb{R}^2$ [7]; deformation spaces of non-uniform hyperbolic lattices in larger Lie groups [15][16]; and the deformation-parameter spaces $R(\Gamma;G;H)$ of discontinuous groups on homogeneous spaces $G/H$ (Clifford–Klein forms), for which open problems are surveyed in [17]. For lattices in Lie groups of general type (nontrivial semisimple Levi factor together with nontrivial solvable radical — the radical being the largest connected solvable normal subgroup of $G$, a Levi factor a connected semisimple subgroup complementary to it), the structure theory of such lattices and their Mostow-type deformations (deformations of a lattice obtained from deformations of the ambient Lie group) is an active topic [14].

Problem Statement

Setup (self-contained). Let $\Gamma$ be a finitely generated discrete group and let $G$ be a connected Lie group. Let $\Re(\Gamma,G)=\mathrm{Hom}(\Gamma,G)$ be the set of group homomorphisms $\Gamma\to G$ with the topology of pointwise convergence; for a finite generating tuple $S=(\gamma_1,\dots,\gamma_k)$ of $\Gamma$, $\Re(\Gamma,G)$ identifies with the subset of $G^k$ defined by the relator equations of a finite presentation of $\Gamma$, a real-analytic subset whose germ-type is independent of the chosen presentation. Let $\Re_0(\Gamma,G)\subseteq\Re(\Gamma,G)$ be the subset of representations $r$ such that: (i) $r$ is injective; (ii) $r(\Gamma)$ is a discrete subgroup of $G$; (iii) the right-coset space $G/r(\Gamma)$ is compact. By Weil's theorem, $\Re_0(\Gamma,G)$ is open in $\Re(\Gamma,G)$. Say that a connected component $C$ of $\Re_0(\Gamma,G)$ is a real-analytic manifold if every point $r\in C$ admits a neighborhood $U$ in the ambient $G^k$ such that $U\cap\Re(\Gamma,G)$ is a real-analytic submanifold germ of $G^k$ (locally the graph of an analytic map over an open subset of a fixed $\mathbb{R}^d$, of locally constant dimension along $C$); in particular $C$ is locally Euclidean in the classical topology. Note that, because $\Re_0(\Gamma,G)$ is open in $\Re(\Gamma,G)$ and real-analytic sets are locally connected and have locally finitely many connected components, the components of $\Re_0(\Gamma,G)$ are real-analytic manifolds in this sense if and only if $\Re(\Gamma,G)$ is a manifold germ at every point of $\Re_0(\Gamma,G)$ — the two formulations of the question are equivalent.

Problem. Determine the pairs $(\Gamma,G)$, with $\Gamma$ finitely generated, $G$ connected, and $\Re_0(\Gamma,G)\neq\varnothing$, for which every connected component of $\Re_0(\Gamma,G)$ is a real-analytic manifold. The general characterization — for each pair, whether every one of its components is a manifold — is target (c) below; its negative side is witnessed, for a given pair $(\Gamma,G)$, by an explicit representation $r\in\Re_0(\Gamma,G)$ together with a local analysis of the germ of $\Re(\Gamma,G)$ at $r$ showing that the component through $r$ is not locally Euclidean. Targets (a) and (b) are the currently open sub-cases:

(a) Solvable targets: $G$ a solvable connected Lie group admitting a lattice $\Gamma$, where $G$ is not assumed simply connected — extend or delimit Wang's identification of the component of the inclusion with an automorphism-group orbit; decide whether each component of $\Re_0(\Gamma,G)$ (equivalently, each component of the space of lattice embeddings $X(\Gamma,G)$) is a manifold, including the cases where $X(\Gamma,G)$ has infinitely many connected components.

(b) Mixed-type targets: $G$ a connected Lie group of mixed type, i.e., with both a nontrivial semisimple Levi factor and a nontrivial solvable radical — the radical being the largest connected solvable normal subgroup of $G$, and a Levi factor a connected semisimple subgroup complementary to the radical (compact factors allowed), and $\Gamma$ a group admitting a faithful representation into $G$ with discrete cocompact image — decide the manifold question for components of $\Re_0(\Gamma,G)$.

(c) General classification: a characterization, in terms of structural data of the pair $(\Gamma,G)$, of exactly when the components of $\Re_0(\Gamma,G)$ are real-analytic manifolds.

Known cases: the Fuchsian case $\Gamma=\pi_1\Sigma_g$ (the fundamental group of a closed orientable surface of genus $g\ge 2$), $G=PSL(2,\mathbb{R})$ (the orientation-preserving isometry group of the hyperbolic plane), where the components are manifolds of Teichmüller type (modeled on the classical deformation space of marked conformal structures on the surface); lattices in simply connected solvable $G$ (Wang); pairs where the relevant lattice embedding is locally rigid (component reduces near it to a single conjugation orbit). A companion analysis of the conjugation quotient (the deformation space of marked compact quotients) — deciding when its strata are manifolds or orbifolds — does not by itself resolve the component question for $\Re_0(\Gamma,G)$.

References

1. André Weil, 'On discrete subgroups of Lie groups', Annals of Mathematics (2) 72 (1960), no. 2, 369–384. DOI: 10.2307/1970140. https://doi.org/10.2307/1970140.
2. André Weil, 'On discrete subgroups of Lie groups (II)', Annals of Mathematics (2) 75 (1962), no. 3, 578–602. DOI: 10.2307/1970212. https://doi.org/10.2307/1970212
3. Hsien-Chung Wang, 'On the Deformations of Lattice in a Lie Group', American Journal of Mathematics 85 (1963), no. 2, 189–212. DOI: 10.2307/2373209. https://doi.org/10.2307/2373209
4. M. S. Raghunathan, 'Discrete Subgroups of Lie Groups', Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 68, Springer-Verlag, New York–Heidelberg, 1972. DOI: 10.1007/978-3-642-86426-1. https://doi.org/10.1007/978-3-642-86426-1 (Chapter on deformations; Weil's openness appears as Theorem 6.19).
5. William M. Goldman, 'Topological components of spaces of representations', Inventiones Mathematicae 93 (1988), no. 3, 557–607. DOI: 10.1007/BF01410200. https://doi.org/10.1007/bf01410200
6. Alexander Lubotzky and Andy R. Magid, 'Varieties of representations of finitely generated groups', Memoirs of the American Mathematical Society 58 (1985), no. 336. DOI: 10.1090/memo/0336. https://doi.org/10.1090/memo/0336
7. Oliver Baues and William M. Goldman, 'Is the deformation space of complete affine structures on the 2-torus smooth?', Contemporary Mathematics 389 (2005), 69–89. DOI: 10.1090/conm/389/07272. https://doi.org/10.1090/conm/389/07272 (also arXiv:math/0401257, https://arxiv.org/abs/math/0401257)
8. Oliver Baues and Benjamin Klopsch, 'Deformations and rigidity of lattices in solvable Lie groups', Journal of Topology 6 (2013), no. 4, 823–856. DOI: 10.1112/jtopol/jtt016. https://doi.org/10.1112/jtopol/jtt016 (also arXiv:1111.5589, https://arxiv.org/abs/1111.5589)
9. William M. Goldman, 'Locally Homogeneous Geometric Manifolds', Proceedings of the International Congress of Mathematicians (Hyderabad, 2010), Vol. II, Hindustan Book Agency, 2011, 717–744. DOI: 10.1142/9789814324359_0071. https://doi.org/10.1142/9789814324359_0071 (also arXiv:1003.2759, https://arxiv.org/abs/1003.2759)
10. Nicolas Bergeron and Tsachik Gelander, 'A Note on Local Rigidity', Geometriae Dedicata 107 (2004), 111–131. DOI: 10.1023/B:GEOM.0000049122.75284.06. https://doi.org/10.1023/B:GEOM.0000049122.75284.06 (also arXiv:1702.00342, https://arxiv.org/abs/1702.00342)
11. Tsachik Gelander and Arie Levit, 'Local rigidity of uniform lattices', Commentarii Mathematici Helvetici 93 (2018), no. 4, 781–827. DOI: 10.4171/CMH/450. https://doi.org/10.4171/cmh/450 (also arXiv:1605.01693, https://arxiv.org/abs/1605.01693)
12. G. A. Margulis and G. A. Soifer, 'Discreteness of deformations of cocompact discrete subgroups', Contemporary Mathematics 772 (2021), 241–247. DOI: 10.1090/conm/772/15492. https://doi.org/10.1090/conm/772/15492
13. Yohei Komori, 'Semialgebraic Description of Teichmüller Space', Publications of the Research Institute for Mathematical Sciences 33 (1997), no. 4, 527–571. DOI: 10.2977/prims/1195145147. https://doi.org/10.2977/prims/1195145147
14. V. V. Gorbatsevich, 'On lattices in Lie groups of general type and some applications', Izvestiya: Mathematics 89 (2025), no. 4, 681–700. DOI: 10.4213/im9643e. https://doi.org/10.4213/im9643e (Russian original: Izv. RAN. Ser. Mat. 89:4 (2025), 32–53; full text at https://www.mathnet.ru/eng/im9643)
15. Sungwoon Kim and Inkang Kim, 'On deformation spaces of nonuniform hyperbolic lattices', Mathematical Proceedings of the Cambridge Philosophical Society 161 (2016), no. 2, 283–303. DOI: 10.1017/S0305004116000293. https://doi.org/10.1017/s0305004116000293
16. Antonin Guilloux and Inkang Kim, 'Deformation Spaces of Discrete Groups of SU(2,1) in Quaternionic Hyperbolic Plane: A Case Study', Experimental Mathematics 30 (2019), no. 4, 453–458. DOI: 10.1080/10586458.2018.1559776. https://doi.org/10.1080/10586458.2018.1559776
17. Ali Baklouti and Imed Kedim, 'Open Problems in Deformation Theory of Discontinuous Groups Acting on Homogeneous Spaces', International Journal of Open Problems in Computer Science and Mathematics 6 (2013), no. 1, 115–131. DOI: 10.12816/0006162. https://doi.org/10.12816/0006162
18. Howard Garland and M. S. Raghunathan, 'Fundamental domains for lattices in (R-)rank 1 semisimple Lie groups', Annals of Mathematics (2) 92 (1970), no. 2, 279–326. DOI: 10.2307/1970838. https://doi.org/10.2307/1970838
19. G. A. Margulis, 'Discrete Subgroups of Semisimple Lie Groups', Ergebnisse der Mathematik und ihrer Grenzgebiete (3), Band 17, Springer-Verlag, Berlin, 1991. DOI: 10.1007/978-3-642-51445-6. https://doi.org/10.1007/978-3-642-51445-6