ABSTRACT
This research proposes a framework to investigate commensurability rigidity for random walks on non-discrete topological groups, synthesizing techniques from various mathematical fields to establish rigidity results.
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Key findings
Develops a comprehensive framework connecting commensurability with probabilistic invariants of random walks.
Formulates conjectures relating spectral properties, Poisson boundaries, and stationary measures to the algebraic structure of commensurators.
Outlines a detailed methodology for investigating rigidity phenomena.
Includes a thorough validation plan with baseline comparisons, ablation studies, and risk analysis.
Limitations & open questions
The non-discrete setting presents unique challenges that require further exploration.