Expansion in simple groups

Abstract

Two short seminal papers of Margulis used Kazhdan's property $(T)$ to give, on the one hand, explicit constructions of expander graphs, and to prove, on the other hand, the uniqueness of some invariant means on compact simple Lie groups. These papers opened a rich line of research on expansion and spectral gap phenomena in finite and compact simple groups. In this paper we survey the history of this area and point out a number of problems which are still open.

arXiv commentsurvey paper, 22 pages

References in corpus (9)

• Expansion of Random Graphs: New Proofs, New Results
• Random subgroups of linear groups are free
• Super-Golden-Gates for PU(2)
• Growth in solvable subgroups of GL_r(Z/pZ)
• Dynamics of Aut(Fn) Actions on Group Presentations and Representations
• Ramanujan Graphs and Digraphs
• Complexity of strong approximation on the sphere
• Borelian subgroups of simple Lie groups
• Quantum Chaos on random Cayley graphs of ${\mathrm{SL}}_2[\mathbb{Z}/p\mathbb{Z}]$

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• Word Measures on Symmetric Groups
• Ramanujan complexes and Golden Gates in PU(3)
• Minimal and nearly minimal measure expansions in connected unimodular groups
• Cutoff on Ramanujan complexes and classical groups
• Certifying Numerical Decompositions of Compact Group Representations