# Symmetries in CR complexity theory

## TL;DR

This paper introduces the Hermitian-invariant group of rational maps between complex balls, explores its properties, and establishes connections between the group's structure and the geometric and algebraic features of the maps.

## Contribution

It defines the Hermitian-invariant group for rational maps, introduces concepts of essential map and source rank, and characterizes maps via their symmetry groups, advancing CR complexity theory.

## Key findings

- Finite subgroups correspond to rational proper maps.
- Non-compactness of the group indicates totally geodesic embeddings.
- Presence of an n-torus in the group characterizes monomial maps.

## Abstract

We introduce the Hermitian-invariant group $\Gamma_f$ 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 $\Gamma_f$ is non-compact if and only if $f$ is a totally geodesic embedding. We show that $\Gamma_f$ contains an $n$-torus if and only if $f$ is equivalent to a monomial map. We show that $\Gamma_f$ 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 $\Gamma_f$ (when the target is a generalized ball) to contain automorphisms that move the origin.