# The Lie group of vertical bisections of a regular Lie groupoid

## TL;DR

This paper constructs an infinite-dimensional Lie group structure on the vertical bisections of a regular Lie groupoid, identifying its Lie algebra and discussing regularity, generalizing known gauge group structures.

## Contribution

It introduces a Lie group structure for vertical bisections of regular Lie groupoids and explores their algebraic and regularity properties, extending previous gauge group results.

## Key findings

- Established Lie group structure on vertical bisections
- Identified the Lie algebra of these groups
- Discussed regularity properties in the sense of Milnor

## Abstract

In this note we construct an infinite-dimensional Lie group structure on the group of vertical bisections of a regular Lie groupoid. We then identify the Lie algebra of this group and discuss regularity properties (in the sense of Milnor) for these Lie groups. If the groupoid is locally trivial, i.e. a gauge groupoid, the vertical bisections coincide with the gauge group of the underlying bundle. Hence the construction recovers the well known Lie group structure of the gauge groups. To establish the Lie theoretic properties of the vertical bisections of a Lie groupoid over a non-compact base, we need to generalise the Lie theoretic treatment of Lie groups of bisections for Lie groupoids over non-compact bases.