TY - JOUR

T1 - Logarithmic girth expander graphs of SLn(Fp)

AU - Arzhantseva, Goulnara

AU - Biswas, Arindam

N1 - Publisher Copyright: © 2022, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.

PY - 2022

Y1 - 2022

N2 - We provide an explicit construction of finite 4-regular graphs (Γk)k∈N with girthΓk→∞ as k→ ∞ and diamΓkgirthΓk⩽D for some D> 0 and all k∈ N. For each fixed dimension n⩾ 2 , we find a pair of matrices in SLn(Z) such that (i) they generate a free subgroup, (ii) their reductions modp generate SLn(Fp) for all sufficiently large primes p, (iii) the corresponding Cayley graphs of SLn(Fp) have girth at least cnlog p for some cn> 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 SLn(Fp) 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.

AB - We provide an explicit construction of finite 4-regular graphs (Γk)k∈N with girthΓk→∞ as k→ ∞ and diamΓkgirthΓk⩽D for some D> 0 and all k∈ N. For each fixed dimension n⩾ 2 , we find a pair of matrices in SLn(Z) such that (i) they generate a free subgroup, (ii) their reductions modp generate SLn(Fp) for all sufficiently large primes p, (iii) the corresponding Cayley graphs of SLn(Fp) have girth at least cnlog p for some cn> 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 SLn(Fp) 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.

KW - Coarse embedding

KW - Diameter

KW - Expander graphs

KW - Large girth graphs

KW - Special linear group

KW - Thin matrix group

UR - http://www.scopus.com/inward/record.url?scp=85130318443&partnerID=8YFLogxK

U2 - 10.1007/s10801-022-01128-z

DO - 10.1007/s10801-022-01128-z

M3 - Journal article

AN - SCOPUS:85130318443

VL - 56

SP - 691

EP - 723

JO - Journal of Algebraic Combinatorics

JF - Journal of Algebraic Combinatorics

SN - 0925-9899

ER -