## Abstract

We introduce the Hermitian-invariant group Γ
_{f} of a proper rational map f between the unit ball in complex Euclidean space and a generalized ball in a space of typically higher dimension. We use properties of the groups to define the crucial new concepts of essential map and the source rank of a map. We prove that every finite subgroup of the source automorphism group is the Hermitian-invariant group of some rational proper map between balls. We prove that Γ
_{f} is non-compact if and only if f is a totally geodesic embedding. We show that Γ
_{f} contains an n-torus if and only if f is equivalent to a monomial map. We show that Γ
_{f} contains a maximal compact subgroup if and only if f is equivalent to the juxtaposition of tensor powers. We also establish a monotonicity result; the group, after intersecting with the unitary group, does not decrease when a tensor product operation is applied to a polynomial proper map. We give a necessary condition for Γ
_{f} (when the target is a generalized ball) to contain automorphisms that move the origin.

Original language | English (US) |
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Pages (from-to) | 590-627 |

Number of pages | 38 |

Journal | Advances in Mathematics |

Volume | 313 |

DOIs | |

State | Published - Jun 20 2017 |

## Keywords

- Automorphism groups
- CR complexity
- Group-invariant CR maps
- Hermitian forms
- Proper holomorphic mappings
- Unitary transformations

## ASJC Scopus subject areas

- Mathematics(all)