Rational Maps of Balls and their Associated Groups
Dusty Grundmeier, Jiří Lebl
TL;DR
This work investigates invariant groups associated to proper rational maps between balls by placing maps in a canonical normal form and analyzing how unitary symmetries constrain these maps. It shows that, in normal form, key groups such as $\Gamma_f$, $G_f$, $D_f$, $\Sigma_f$, and $\Delta^{(a,b)}_f$ are subgroups of unitary groups and are determined by the invariants of the underlying Hermitian form $|g|^2-\|p\|^2$. A central result is that $\Gamma_f$ is defined by a single invariant polynomial, and conversely any subgroup of $U(n)$ defined by invariant polynomials can be realized as $\Gamma_f$ for some polynomial (or rational) ball map, enabling explicit constructions of maps with prescribed symmetries, including degree-$4$ examples with controlled denominators. The paper also provides constructive methods to realize given denominators and to produce maps with specific symmetry groups, highlighting how symmetry constraints influence map degree and evenness of numerator/denominator. These results enrich the toolkit for understanding the interaction between holomorphic maps and their group symmetries in several complex variables.
Abstract
Given a proper, rational map of balls, D'Angelo and Xiao introduced five natural groups encoding properties of the map. We study these groups using a recently discovered normal form for rational maps of balls. Using this normal form, we also provide several new groups associated to the map.