# Infinite nonabelian corks

## TL;DR

This paper constructs new types of corks for infinite nonabelian groups, expanding the understanding of exotic smooth structures on 4-manifolds and answering a question posed by Tange.

## Contribution

It introduces the first examples of $G$-corks for infinite nonabelian groups and combines previous results to achieve this construction.

## Key findings

- Constructed $G$-corks for any extension of $\\mathbb Z^m$ by finite subgroups of $SO(4)$.
- Provided weakly equivariant $G$-corks for extensions by finite solvable groups.
- Applied Gompf's results to produce exotic $\\mathbb R^4$'s with diffeotopy groups containing all poly-cyclic groups.

## Abstract

We construct $G$-corks for any extension $G$ of $\mathbb Z^m$ by any finite subgroup of $\mathrm{SO}(4)$ and weakly equivariant $G$-corks for any extension $G$ of $\mathbb Z^m$ by any finite solvable group. In particular, this is the first example of $G$-corks for an infinite nonabelian group $G$ and answers a question by Tange. The construction is a combination of previous results by Auckly-Kim-Melvin-Ruberman, Gompf, and Tange. Using Gompf's results about exotic $\mathbb R^4$'s, we give an application to construct exotic $\mathbb R^4$'s whose diffeotopy group contains all poly-cyclic groups.