On systolic growth of Lie groups

Introduced by Gromov in the nineties, the systolic growth of a
finitely generated group maps $n$ to the smallest index of a finite
index subgroup meeting the $n$-ball only in the identity singleton.
This function is one measure of residual finiteness. It extends to
compactly generated locally compact groups, replacing "finite index"
with "cocompact lattice" in the definition.

It grows as least as fast as the word growth, and with Bou-Rabee we
showed that the growth is exponential for linear groups of exponential
growth.

For finitely generated groups with polynomial growth, the systolic
growth is also polynomially bounded, but possibly with worse exponent.
For a lattice $\Gamma$ in a simply connected nilpotent Lie group $G$,
we show the following: $\Gamma$ has systolic growth asymptotically
equivalent to the word growth if and only if the same holds for $G$,
if and only the Lie algebra of $G$ admits a Carnot grading. In some
non-Carnot cases, we provide an estimate of the systolic growth; for
instance in one case it grows as $n^k$ for some non-integer $k$.