Logarithmic girth expander graphs of SL_n( F_p) · arXivDesk
1803.09229Mar 25, 2018Title and content updated to reflect published version. Previous title: "Large girth graphs with bounded diameter-by-girth ratio"
We provide an explicit construction of finite 4-regular graphs (Γk)k∈N with girthΓk→∞
Nearby in the stack
as
k→∞
and
girthΓkdiamΓk⩽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
cnlogp
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(logp)
. 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.