On local sharply n-transitive groups

TL;DR

This paper develops a theory of local sharply n-transitive groups, generalizing classical group actions on manifolds by introducing local groups and their algebraic reductions, applicable for any n.

Contribution

It introduces the concept of local sharply n-transitive groups and local n-pseudofields, extending the classical theory to local topological groups and providing algebraic reductions.

Findings

Local sharply n-transitive groups exist for any n.

These groups can be reduced to local n-pseudofields, similar to Lie groups and Lie algebras.

Boundedly sharply n-transitive groups are Lie groups, enabling additional analysis methods.

Abstract

The paper is devoted to generalizations of actions of topological groups on manifolds. Instead of a topological group, we consider a local topological group generalizing the notion of a~germ or a~neighborhood in a topological group. The notion of an action of a local group on a topological space is introduced. The paper constructs the theory of local sharply $n$-transitive groups and local $n$-pseudofields. Local sharply $n$-transitive groups are reduced to simpler algebraic objects -- local $n$-pseudofields, similarly to the way Lie groups are reduced to Lie algebras, and sharply two-transitive groups, are reduced to neardomains. This can be useful, since, opposite to locally compact and connected sharply $n$-transitive groups, which are absent for $n > 3$, local sharply $n$-transitive groups exist for any $n$, for example, the group $GL_n(\mathbb{R})$. Being boundedly sharply $n$-transitive, the groups under consideration are also Lie groups, which gives extra methods for their study.