Polynomial mappings of groups
Let \(\phi\) be a mapping of a group \(G\) to a group \(F\) and let \(x\in G\). Define the operator \(D_x\) by \((D_x\phi)(g)=\phi(g)\phi(gx)\), \(g\in G\). A mapping \(\phi\colon G\to F\) is called a polynomial of degree \(\leq d\) if \(D_{x_1}D_{x_2}\cdots D_{x_{d+1}}\phi\cong 1_F\) for any \(x_1,\dots,x_{d+1}\in G\). The basic part of the paper is section 3. Here the author studies the polynomial mappings to a nilpotent group. We denote some results of this section. Theorem 3.2. Let \(G\) be a group and \(F\) a nilpotent group. Then the polynomial mappings \(G\to F\) form a group with respect to the elementwise multiplication. Corollary 3.7. Let \(G\) be a nilpotent group. Then the operations of multiplication \((x,y)\to xy\), \(x,y\in G\) and of raising to a power \((x,n)\to x^n\), \(x\in G\) are polynomial mappings. Proposition 3.20. Let \(F\) be a soluble group, \(G\) a group and \(\phi\colon G\to F\) be a polynomial mapping. Then \(\phi\) is representable as the composition of a homomorphism \(\pi\) of \(G\) onto a soluble group \(H\) and a polynomial mapping \(\psi\colon H\to F\). Proposition 3.21. Let \(F\) be a nilpotent group, \(G\) a group and \(\phi\colon G\to F\) be a polynomial mapping. Then \(\phi\) is representable as the composition of a homomorphism \(\pi\) of \(G\) onto a nilpotent group \(H\) and a polynomial mapping \(\psi\colon H\to F\). The author applies the results obtained to prove the existence of the compact/weak mixing decomposition of a Hilbert space under a unitary polynomial action of a finitely generated nilpotent group.
scientific article; zbMATH DE number 1760704