Building Monotone Expanders

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Constructs explicit constant-degree monotone expanders by first building a continuous monotone expander on [0,1] from a finite set of SL(2,R) Möbius maps (take words of fixed length in three matrices), then discretizing intervals to get finite bipartite graphs so they are constant-page as well. Proof uses three concrete steps: an effective Tits alternative to get large girth, a metric-entropy product-growth estimate from additive combinatorics, and mixing from the 3‑transitive Möbius action. Result: monotone and dimension expanders exist, with enormous degrees and tiny explicit expansion constants.

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1. 0:00Intro
2. 0:55(bipartite) vertex expansion
3. 1:44background
4. 3:52how simple can expanders be?
5. 18:40a monotone expander
6. 19:23(a) Schreier diagrams
7. 20:47(a) an example
8. 22:25(b) continuous monotone expanders
9. 28:55(c) monotone expanders: concluding
10. 29:46(c) monotonicity
11. 30:20(d) a three-step proof: chess game [Sarnak]
12. 32:50(d) opening: large girth
13. 37:21(d) middle-game: product-growth
14. 39:21d endgame: mixing property an example