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Logarithmic girth expander graphs of $$SL n({\mathbb {F}} p)$$ by Goulnara Arzhantseva; Arindam Biswas is a Mathematics article available to read on EtoBox.

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We provide an explicit construction of finite 4-regular graphs ( k ) k∈N with girth k → ∞ as k → ∞ and diam k girth k ≤ D for some D > 0 and all k ∈ N. For each fixed dimension n ≥ 2, we find a pair of matrices in SL n (Z) such that (i) they generate a free subgroup, (ii) their reductions mod p generate SL n (F p ) for all sufficiently large primes p, (iii) the corresponding Cayley graphs of SL n (F p ) have girth at least c n log p for some c n > 0. Relying on growth results (with no use of expansion properties of the involved graphs), we observe that the diameter of those Cayley graphs is at most O(log p). This gives infinite sequences of finite 4-regular Cayley graphs of SL n (F p ) as p → ∞ with large girth and bounded diameter-by-girth ratio. These are the first explicit examples in all dimensions n ≥ 2 (all prior examples were in n = 2). Moreover, they happen to be expanders. Together with Margulis' and Lubotzky-Phillips-Sarnak's classical constructions, these new graphs are the only known explicit logarithmic girth Cayley graph expanders.

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Author

Goulnara Arzhantseva; Arindam Biswas

Publisher

Springer Science and Business Media LLC

Published

2022

Language

EN

Field

Mathematics (Physical Sciences)