Periodic Reporting for period 4 - DiGGeS (Discrete Groups and Geometric Structures)
Période du rapport: 2022-03-01 au 2023-12-31
Discrete subgroups of Lie groups are fundamental mathematical objects encoding symmetries of geometric spaces. They lie at the intersection of geometry and group theory with various other fields of mathematics, including Lie theory, differential equations, complex analysis, ergodic theory, representation theory, algebraic geometry, number theory, and mathematical physics.
To any noncompact semisimple Lie group is associated a positive integer called the real rank. In real rank one, important classes of discrete subgroups of semisimple Lie groups are known for their good geometric, topological, and dynamical properties, such as convex cocompact or geometrically finite subgroups. On the other hand, in higher real rank, discrete subgroups beyond lattices remain more mysterious: the goal of the project was to understand them better.
More precisely, our objective was to identify and investigate various significant classes of discrete subgroups of semisimple Lie groups in higher real rank, making links with geometric structures on manifolds. For this we studied the geometry and dynamics of actions of discrete groups on various homogeneous spaces: flag varieties, seen "at infinity" of Riemannian symmetric spaces, but also pseudo-Riemannian symmetric spaces such as the pseudo-Riemannian analogues of real hyperbolic space. We used such geometric and dynamical information to construct properly discontinuous actions on affine spaces and to develop a spectral theory of the pseudo-Riemannian Laplacian on certain locally homogeneous spaces.
To any noncompact semisimple Lie group is associated a positive integer called the real rank. In real rank one, important classes of discrete subgroups of semisimple Lie groups are known for their good geometric, topological, and dynamical properties, such as convex cocompact or geometrically finite subgroups. On the other hand, in higher real rank, discrete subgroups beyond lattices remain more mysterious: the goal of the project was to understand them better.
More precisely, our objective was to identify and investigate various significant classes of discrete subgroups of semisimple Lie groups in higher real rank, making links with geometric structures on manifolds. For this we studied the geometry and dynamics of actions of discrete groups on various homogeneous spaces: flag varieties, seen "at infinity" of Riemannian symmetric spaces, but also pseudo-Riemannian symmetric spaces such as the pseudo-Riemannian analogues of real hyperbolic space. We used such geometric and dynamical information to construct properly discontinuous actions on affine spaces and to develop a spectral theory of the pseudo-Riemannian Laplacian on certain locally homogeneous spaces.
The work performed is in line with the above objective. It led to 46 papers.
Outside of lattices, an important class of discrete subgroups of higher-rank semisimple Lie groups is the Anosov subgroups, i.e. the images of the Anosov representations of Gromov hyperbolic groups introduced by Labourie in 2006 and further studied by Guichard-Wienhard and many others. These representations, defined by a strong dynamical condition, play a central role in higher Teichmüller theory. The project investigated various aspects of Anosov subgroups and their limit sets. We obtained new characterisations, and proposed a generalisation to semigroups.
Work of Labourie, Guichard-Wienhard, Kapovich-Leeb-Porti and others shows that Anosov subgroups of a higher-rank semisimple Lie group G have many similarities with convex cocompact subgroups of rank-one Lie groups. However, from a geometric point of view, Anosov subgroups are not convex cocompact in an obvious way: they do not satisfy any convex cocompactness property in the Riemannian symmetric space of G (by work of Kleiner-Leeb and Quint).
An important aspect of the project was to introduce and investigate various notions of convex cocompactness in other settings, namely in pseudo-Riemannian hyperbolic spaces and in real projective spaces. We obtained characterisations of Anosov subgroups as convex cocompact subgroups in these settings. These geometric intepretations of Anosov subgroups led to the construction of many new examples, using the geometry of linear reflection groups à la Vinberg.
Based on a geometric study of convex cocompactness in pseudo-hyperbolic spaces, the project included the construction of new examples of what could be called higher higher Teichmüller spaces: namely, connected components of G-character varieties of pi_1(M) consisting entirely of injective and discrete representations, where G is a higher-rank semisimple Lie group and M a higher-dimensional manifold.
While convex cocompact groups in rank one are always hyperbolic in the sense of Gromov, an interesting part of the project was to develop a general notion of convex cocompactness in higher rank involving discrete groups that are not necessarily Gromov hyperbolic anymore. Still, we showed that a number of good properties from rank one remain in the higher-rank setting, even in the absence of hyperbolicity. Examples of nonhyperbolic convex cocompact groups include groups dividing a nonstrictly convex domain in projective space, as studied in particular by Benoist, and deformations of those in higher dimension. The project included the construction of many other examples using representations of Coxeter groups as linear reflection groups à la Vinberg; we actually gave a full description of all possible examples in this setting. Various dynamical aspects of convex cocompact groups in higher rank were also studied as part of the project, leading to a Patterson-Sullivan theory in this setting and to various counting results.
Other generalisations of Anosov subgroups were investigated as part of the project, including notions of relatively Anosov subgroups developed by Kapovich-Leeb and Zhu, as well as other discrete subgroups with good dynamical properties.
The project also included a study of properly discontinuous and cocompact actions on pseudo-Riemannian symmetric spaces, or more generally reductive homogeneous spaces. In particular, the PI and Tholozan proved the Sharpness Conjecture for such actions. As a consequence, for any reductive homogeneous space G/H of corank one, we characterised the discrete subgroups of G acting properly discontinuously and cocompactly on G/H as Anosov subgroups. Applications include the non-existence of compact quotients of certain homogeneous spaces such as SL(2n)/SL(2n-1).
A better understanding of various classes of discrete subgroups of semisimple Lie groups led, as part of the project, to progress on the study of properly discontinuous affine actions of discrete groups on R^n. In particular, we constructed properly discontinuous affine actions by any right-angled Coxeter group. We also studied the possible Zariski closures of nonvirtually solvable discrete groups with proper affine actions on R^n. Properness criteria were obtained in various settings.
In a different direction, the project developed a spectral theory of the Laplacian on certain quotients of pseudo-Riemannian symmetric spaces, in relation with representation theory. Several of the aforementioned results, such as the Sharpness Conjecture, have applications in this setting.
Outside of lattices, an important class of discrete subgroups of higher-rank semisimple Lie groups is the Anosov subgroups, i.e. the images of the Anosov representations of Gromov hyperbolic groups introduced by Labourie in 2006 and further studied by Guichard-Wienhard and many others. These representations, defined by a strong dynamical condition, play a central role in higher Teichmüller theory. The project investigated various aspects of Anosov subgroups and their limit sets. We obtained new characterisations, and proposed a generalisation to semigroups.
Work of Labourie, Guichard-Wienhard, Kapovich-Leeb-Porti and others shows that Anosov subgroups of a higher-rank semisimple Lie group G have many similarities with convex cocompact subgroups of rank-one Lie groups. However, from a geometric point of view, Anosov subgroups are not convex cocompact in an obvious way: they do not satisfy any convex cocompactness property in the Riemannian symmetric space of G (by work of Kleiner-Leeb and Quint).
An important aspect of the project was to introduce and investigate various notions of convex cocompactness in other settings, namely in pseudo-Riemannian hyperbolic spaces and in real projective spaces. We obtained characterisations of Anosov subgroups as convex cocompact subgroups in these settings. These geometric intepretations of Anosov subgroups led to the construction of many new examples, using the geometry of linear reflection groups à la Vinberg.
Based on a geometric study of convex cocompactness in pseudo-hyperbolic spaces, the project included the construction of new examples of what could be called higher higher Teichmüller spaces: namely, connected components of G-character varieties of pi_1(M) consisting entirely of injective and discrete representations, where G is a higher-rank semisimple Lie group and M a higher-dimensional manifold.
While convex cocompact groups in rank one are always hyperbolic in the sense of Gromov, an interesting part of the project was to develop a general notion of convex cocompactness in higher rank involving discrete groups that are not necessarily Gromov hyperbolic anymore. Still, we showed that a number of good properties from rank one remain in the higher-rank setting, even in the absence of hyperbolicity. Examples of nonhyperbolic convex cocompact groups include groups dividing a nonstrictly convex domain in projective space, as studied in particular by Benoist, and deformations of those in higher dimension. The project included the construction of many other examples using representations of Coxeter groups as linear reflection groups à la Vinberg; we actually gave a full description of all possible examples in this setting. Various dynamical aspects of convex cocompact groups in higher rank were also studied as part of the project, leading to a Patterson-Sullivan theory in this setting and to various counting results.
Other generalisations of Anosov subgroups were investigated as part of the project, including notions of relatively Anosov subgroups developed by Kapovich-Leeb and Zhu, as well as other discrete subgroups with good dynamical properties.
The project also included a study of properly discontinuous and cocompact actions on pseudo-Riemannian symmetric spaces, or more generally reductive homogeneous spaces. In particular, the PI and Tholozan proved the Sharpness Conjecture for such actions. As a consequence, for any reductive homogeneous space G/H of corank one, we characterised the discrete subgroups of G acting properly discontinuously and cocompactly on G/H as Anosov subgroups. Applications include the non-existence of compact quotients of certain homogeneous spaces such as SL(2n)/SL(2n-1).
A better understanding of various classes of discrete subgroups of semisimple Lie groups led, as part of the project, to progress on the study of properly discontinuous affine actions of discrete groups on R^n. In particular, we constructed properly discontinuous affine actions by any right-angled Coxeter group. We also studied the possible Zariski closures of nonvirtually solvable discrete groups with proper affine actions on R^n. Properness criteria were obtained in various settings.
In a different direction, the project developed a spectral theory of the Laplacian on certain quotients of pseudo-Riemannian symmetric spaces, in relation with representation theory. Several of the aforementioned results, such as the Sharpness Conjecture, have applications in this setting.
The ERC project successfully led to a better understanding of discrete subgroups of higher-rank semisimple Lie groups, beyond the state of the art, as described above. Two survey articles were written by the PI explaining the details of this progress. The ERC funding contributed to important dissemination of knowledge through the organisation of a monthly "Geometry and discrete groups" seminar at the IHES, of numerous conferences and workshops, and of two summer/winter schools for young researchers. The project has opened up new perspectives and will continue to stimulate new research in the upcoming years.