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Homological dimension of discrete subgroups in higher rank Lie groups

Chris Connell, D. B. McReynolds, Shi Wang

TL;DR

The paper addresses the problem of establishing a homological dimension gap for infinite-volume, torsion-free discrete subgroups $\Gamma$ of higher rank Lie groups by sharpening upper bounds on the critical index $j_X(\Gamma)$ for irreducible symmetric spaces $X=G/K$. Building on Patterson–Sullivan theory and the flow framework of CMW23, the authors derive explicit inequalities $j_X(\Gamma) < n+1 - c_r$ with type- and rank-dependent constants $c_r$, achieved by a refined choice of the test functional $\phi$ proportional to $2\rho-\Theta$ and explicit root-data computations. The main outcome is a suite of corollaries giving homological dimension bounds $\mathrm{hd}_R(\Gamma) < n - c_r$ for various real forms, under the assumption that the injectivity radius of $\Gamma\backslash X$ is uniformly bounded away from zero; several exceptional low-rank cases are automatic. The results unify real split and non-split cases across all irreducible types $(A_r,B_r,C_r,D_r,(BC)_r)$ and discuss sharpness and potential removal of the injectivity radius condition, suggesting the optimal bound $\mathrm{hd}_R(\Gamma) \le n - r$ in general.

Abstract

In this short note, we improve on a recent result by the authors. We show that infinite volume torsion free discrete subgroups of higher rank Lie groups have homological dimension gap at least one-eighth of the real rank, provided the injectivity radius of the quotient manifold is uniformly bounded from zero.

Homological dimension of discrete subgroups in higher rank Lie groups

TL;DR

The paper addresses the problem of establishing a homological dimension gap for infinite-volume, torsion-free discrete subgroups of higher rank Lie groups by sharpening upper bounds on the critical index for irreducible symmetric spaces . Building on Patterson–Sullivan theory and the flow framework of CMW23, the authors derive explicit inequalities with type- and rank-dependent constants , achieved by a refined choice of the test functional proportional to and explicit root-data computations. The main outcome is a suite of corollaries giving homological dimension bounds for various real forms, under the assumption that the injectivity radius of is uniformly bounded away from zero; several exceptional low-rank cases are automatic. The results unify real split and non-split cases across all irreducible types and discuss sharpness and potential removal of the injectivity radius condition, suggesting the optimal bound in general.

Abstract

In this short note, we improve on a recent result by the authors. We show that infinite volume torsion free discrete subgroups of higher rank Lie groups have homological dimension gap at least one-eighth of the real rank, provided the injectivity radius of the quotient manifold is uniformly bounded from zero.

Paper Structure

This paper contains 23 sections, 4 theorems, 81 equations, 6 tables.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['thm:j_X_bound']}--Real split cases
  3. Proof of Theorem \ref{['thm:j_X_bound']}--Non-split cases
  4. Type $A_r$:
  5. $G=\mathop{\mathrm{SL}}\nolimits_{r+1}(\mathbb C)$:
  6. $G=\mathop{\mathrm{SU}}\nolimits^*_{4r+2}$:
  7. $G=E_6^{-26}$:
  8. Type $B_r$:
  9. $G=\mathop{\mathrm{SO}}\nolimits_{r,r+k}$:
  10. $G=\mathop{\mathrm{SO}}\nolimits_{2r+1}(\mathbb C)$:
  11. Type $C_r$:
  12. $G=\mathop{\mathrm{SU}}\nolimits_{r,r}$:
  13. $G=\mathop{\mathrm{Sp}}\nolimits_r(\mathbb C)$:
  14. $G=\mathop{\mathrm{SO}}\nolimits^*_{4r}$:
  15. $G=\mathop{\mathrm{Sp}}\nolimits_{r,r}$:
  16. ...and 8 more sections

Key Result

Theorem 1.1

CMW23 If $X$ is a Hadamard space with bounded sectional curvatures, $\Gamma < \mathrm{Isom}(X)$ is a discrete, torsion-free subgroup with $M = X/\Gamma$, and $\mathbb{V}$ is a flat bundle over $M$, then for every $\epsilon > 0$, the homomorphism $i_k:H_k(M_{\varepsilon};\mathbb V)\rightarrow H_k(M;\

Theorems & Definitions (11)

  • Theorem 1.1
  • Definition 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Corollary 1.5
  • proof
  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • ...and 1 more