Solutions to open problems in Neeb's recent survey on infinite-dimensional Lie groups

TL;DR

This paper addresses three open problems in infinite-dimensional Lie groups, demonstrating specific subgroup structures, conditions for direct limits, and a version of Borel's Theorem related to smooth diffeomorphisms.

Contribution

It solves three open problems from Neeb's survey on infinite-dimensional Lie groups and proves a version of Borel's Theorem for smooth diffeomorphisms.

Findings

Existence of a subgroup without an initial Lie subgroup structure.

Pathology does not occur in direct limits of finite-dimensional Lie groups.

Every such direct limit group is a topological group with Lie algebra.

Abstract

We solve three open problems concerning infinite-dimensional Lie groups posed in a recent survey article by K.-H. Neeb: (1) There exists a subgroup of some infinite-dimensional Lie group G which does not admit an initial Lie subgroup structure; (2) The pathology cannot occur if G is a direct limit of an ascending sequence of finite-dimensional Lie groups; (3) Every such direct limit group is a ``topological group with Lie algebra'' in the sense of Hofmann and Morris. Moreover, we prove a version of Borel's Theorem announced in the survey, ensuring the existence of compactly supported smooth diffeomorphisms with given Taylor series around a fixed point p (provided the tangent map at p has positive determinant).