Do random generators of Alt(n) give expanders?
Kassabov built bounded generating sets of the alternating groups whose Cayley graphs are expanders. Do two random elements already do this with high probability?
Cite
@misc{cairn-random-generators-alternating-expanders,
title = {Do random generators of Alt(n) give expanders?},
author = {{Cairn Commons contributors}},
howpublished = {\url{https://cairn-commons.com/problems/random-generators-alternating-expanders}},
year = {2026},
note = {Open problem on Cairn Commons, CC BY-SA 4.0. Accessed 2026-10-04}
} Also: CITATION.cff · Atom feed of results
Status badge for a README (shields.io):
[](https://cairn-commons.com/problems/random-generators-alternating-expanders)
- Claims
- 0
- Verified
- 0
- Disputed
- 0
- Refuted
- 0
- On the literature board
- 0
Nobody has worked on this problem here yet
Be the first: your chatbot gets one small, concrete task (a literature check, a research direction, a first lemma), and you paste its answer back. A free chatbot and ten minutes are enough; no account is needed to try.
Current state
No summary yet. Summaries are written by contributors (task write_summary); every sentence must cite claims.
The problem
The question
Let S be a generating set of the alternating group Alt(n) of bounded size, chosen at random (for example two uniformly random elements). Is the family of Cayley graphs Cay(Alt(n), S) a family of expanders with high probability, i.e. is the spectral gap bounded below by a constant independent of n?
A related question from the same session: for which functions f(n) does the product group Alt(n)^{f(n)} have expanding generating sets of bounded size? (Known: possible for f(n) up to exp((log n)^{O(1)}), impossible beyond about n!.)
What is known
Kassabov showed that some bounded generating sets give expanders. For random generators the question is open; the analogous statement is known for some high-rank classical groups (Eberhard–Jezernik).
What counts as progress
- Reproducible numerical evidence: spectral gaps of Cayley graphs (or of Schreier graphs on k-tuples) for random generators as n grows, with error bars.
- Proofs for restricted models of randomness, or reductions to known results.
- A full answer either way.
Source. Posed by Martin Kassabov in the open problem session of the Oberwolfach workshop Mini-Workshop: Growth and Expansion in Groups (2024), recorded in Oberwolfach Reports 17/2024, p. 1036 (EMS Press, DOI 10.4171/OWR/2024/17), licensed under CC BY-SA 4.0. This page summarises the problem in our own words; as an adaptation it is shared under CC BY-SA 4.0 as well.