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## Symmetric groups and expanders

### Martin Kassabov

**Abstract.**
We construct explicit generating sets $F_n$ and $\tilde F_n$ of the
alternating and the symmetric groups,
which turn the Cayley graphs $\mathcal{C}(\textup{Alt}(n), F_n)$
and $\mathcal{C}(\textup{Sym}(n), \tilde F_n)$ into
a family of bounded degree expanders for all sufficiently large $n$.
These expanders have many applications
in the theory of random walks on groups and in other areas of mathematics.

*Copyright 2005 American Mathematical Society
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#### Article Info

- ERA Amer. Math. Soc.
**11** (2005), pp. 47-56
- Publisher Identifier: S 1079-6762(05)00146-0
- 2000
*Mathematics Subject Classification*. Primary 20B30; Secondary 05C25, 05E15, 20C30, 20F69, 60C05, 68R05, 68R10
*Key words and phrases.* Expanders, symmetric groups,
alternating groups, random permutations, property T, Kazhdan constants
- Received by editors March 16, 2005
- Posted on June 9, 2005
- Communicated by Efim Zelmanov
- Comments (When Available)

**Martin Kassabov**

Department of Mathematics, Cornell University, Ithaca, New York 14853-4201

*E-mail address:* `kassabov@math.cornell.edu`

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