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Effective density of non-degenerate random walks on homogeneous spaces

Wooyeon Kim, Constantin Kogler

Abstract

We prove effective density of random walks on homogeneous spaces, assuming that the underlying measure is supported on matrices generating a dense subgroup and having algebraic entries. The main novelty is an argument passing from high dimension to effective equidistribution in the setting of random walks on homogeneous spaces, exploiting spectral gap of the associated convolution operator.

Effective density of non-degenerate random walks on homogeneous spaces

Abstract

We prove effective density of random walks on homogeneous spaces, assuming that the underlying measure is supported on matrices generating a dense subgroup and having algebraic entries. The main novelty is an argument passing from high dimension to effective equidistribution in the setting of random walks on homogeneous spaces, exploiting spectral gap of the associated convolution operator.
Paper Structure (12 sections, 17 theorems, 96 equations)

This paper contains 12 sections, 17 theorems, 96 equations.

Table of Contents

  1. Introduction
  2. Outline and Notation
  3. Outline of proofs
  4. Notation
  5. Constants
  6. Proof of Theorem \ref{['MainTheorem']}
  7. Quantitative Non-Divergence
  8. High Dimension
  9. Proof of Theorem \ref{['MainTheorem']}
  10. Proof of Theorem \ref{['EffectiveDiameter']} and Theorem \ref{['EffectiveDensity']}
  11. Proof of Theorem \ref{['EffectiveDiameter']}
  12. Proof of Theorem \ref{['EffectiveDensity']}

Key Result

Theorem 1.1

Let $G$ be a connected simple Lie group with finite center, $\Lambda < G$ a lattice and $X = G/\Lambda$. Let $S \subset G$ be a symmetric set generating a dense subgroup of $G$. Assume further that there is a basis of $\mathfrak{g}$ such that $\mathrm{Ad}(S)$ consists of matrices with algebraic entr where the implied constant depends on $\Lambda$ and $S$.

Theorems & Definitions (31)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Proposition 3.1
  • Lemma 3.2
  • proof
  • Proposition 3.3
  • Proposition 3.4
  • ...and 21 more