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On cusp holonomies in strictly convex projective geometry

Balthazar Fléchelles

TL;DR

The paper provides a complete characterization of cusp holonomies in strictly convex and round convex projective geometry by establishing a criterion based on $P_1$-divergence and limit sets, and then extends the generalized cusp framework to include virtually solvable holonomies, yielding non-nilpotent examples. It develops a geometric and analytic toolkit—Hilbert geometry, duality, Vinberg hypersurfaces, and a smoothing lemma—to construct and deform cusp domains while preserving cusp structure. A key contribution is linking cusp holonomies to relatively Anosov representations, clarifying the periphery structure of these representations and enabling new constructions with virtually nilpotent or solvable holonomies. The work also demonstrates the existence of generalized cusps of non-maximal rank, including explicit solvable-holonomy examples, thereby highlighting the broader landscape of cusps in convex projective geometry and its connections to higher-rank dynamics.

Abstract

We give a complete characterization of the holonomies of strictly convex cusps and of round cusps in convex projective geometry. We build families of generalized cusps of non-maximal rank associated to each strictly convex or round cusp. We also extend Ballas-Cooper-Leitner's definition of generalized cusp to allow for virtually solvable fundamental group, and we produce the first such example with non-virtually nilpotent fundamental group. Along with a companion paper, this allows to build strictly convex cusps and generalized cusps whose fundamental group is any finitely generated virtually nilpotent group. This also has interesting consequences for the theory of relatively Anosov representations.

On cusp holonomies in strictly convex projective geometry

TL;DR

The paper provides a complete characterization of cusp holonomies in strictly convex and round convex projective geometry by establishing a criterion based on -divergence and limit sets, and then extends the generalized cusp framework to include virtually solvable holonomies, yielding non-nilpotent examples. It develops a geometric and analytic toolkit—Hilbert geometry, duality, Vinberg hypersurfaces, and a smoothing lemma—to construct and deform cusp domains while preserving cusp structure. A key contribution is linking cusp holonomies to relatively Anosov representations, clarifying the periphery structure of these representations and enabling new constructions with virtually nilpotent or solvable holonomies. The work also demonstrates the existence of generalized cusps of non-maximal rank, including explicit solvable-holonomy examples, thereby highlighting the broader landscape of cusps in convex projective geometry and its connections to higher-rank dynamics.

Abstract

We give a complete characterization of the holonomies of strictly convex cusps and of round cusps in convex projective geometry. We build families of generalized cusps of non-maximal rank associated to each strictly convex or round cusp. We also extend Ballas-Cooper-Leitner's definition of generalized cusp to allow for virtually solvable fundamental group, and we produce the first such example with non-virtually nilpotent fundamental group. Along with a companion paper, this allows to build strictly convex cusps and generalized cusps whose fundamental group is any finitely generated virtually nilpotent group. This also has interesting consequences for the theory of relatively Anosov representations.

Paper Structure

This paper contains 32 sections, 51 theorems, 77 equations.

Table of Contents

  1. Introduction
  2. Strictly convex geometry
  3. Characterization of strictly convex cusp holonomies
  4. Non-elementary cusps and relatively Anosov representations
  5. The non-strictly convex setting: generalized cusps
  6. Organization of the paper
  7. Acknowledgments
  8. Reminders about convex projective geometry
  9. The Hilbert metric
  10. Duality
  11. Classification of automorphisms
  12. Vinberg hypersurfaces
  13. Elementary groups and cusps
  14. Algebraic horospheres
  15. Elementary groups in strictly convex domains
  16. ...and 17 more sections

Key Result

Theorem 1.4

Let $\Gamma$ be a discrete subgroup of $\mathrm{PGL}(n,\mathbb{R})$. Then $\Gamma$ is the holonomy of a strictly convex (resp. round) cusp if and only if $\Gamma$ preserves a properly convex domain, is $P_1$-divergent, and its limit set $\Lambda_1(\Gamma)$ (resp. $\Lambda_{1,n-1}(\Gamma)$) is redu

Theorems & Definitions (130)

  • Definition 1.3
  • Theorem 1.4: see Thm. \ref{['thm:characHoloStrictlyConvCusps']} & \ref{['thm:characHoloRoundCusps']}
  • Theorem 1.5: see flechelles2025unipotentReps
  • Definition 1.6
  • Definition 1.7
  • Definition 1.8
  • Theorem 1.9
  • Theorem 1.10
  • Definition 1.11
  • Theorem 1.12: see Prop. \ref{['prop:geomGenCuspsAreGenCusps']} & Cor. \ref{['cor:genCuspDomainsForVirtNilpGps']}
  • ...and 120 more