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 $k$.

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