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Estimates of the action order of some jet groups

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*Karolina Lubas and Andrzej Zajtz*

**E-mail:** smzajtz@cyf-kr.edu.pl
**Abstract.** As we know, if the general $k$-jet group $G(n,k)$ in
$\R$ acts continuously on a manifold of finite dimension $m>0$ and the
highest $k$-th order derivatives act nontrivially, then $k$ is bounded
by a known function of $m$ and $n$ [4]. In geometric interpretation the
estimate function translates into the order of the natural bundle in which
the group $G(n,k)$ acts on the standard fibre (meant as the space of geometric
objects). On manifolds with extra structure the same question arises for
corresponding subgroups of $G(n,k)$, preserving the structure. In this
note the sharp upper estimates are found for actions of unimodular, homothetic
and symplectic subgroups.

**AMSclassification.** 58G99

**Keywords.** Unimodular, homothetic, symplectic jet groups and Lie
algebras, bounds to codimension of subalgebras, action order