# $\Delta$-Weakly Mixing Subset In Positive Entropy Actions Of A Nilpotent Group

## TL;DR

This paper introduces $\Delta$-weakly mixing subsets for group actions and shows their existence under positive entropy for nilpotent groups, highlighting differences with solvable groups.

## Contribution

It defines $\Delta$-weakly mixing subsets for nilpotent group actions and establishes their existence under positive entropy, contrasting with solvable groups.

## Key findings

- Positive entropy implies $\Delta$-weakly mixing subsets for nilpotent groups.
- Existence of positive entropy actions without $\Delta$-weakly mixing subsets in solvable groups.
- Highlights differences between nilpotent and solvable group actions.

## Abstract

The notion of $\Delta$-weakly mixing subsets is introduced for countable torsion-free discrete group actions. It is shown that for a finitely generated torsion-free discrete nilpotent group action, positive topological entropy implies the existence of $\Delta$-weakly mixing subsets, and while there exists a finitely generated torsion-free discrete solvable group action which has positive topological entropy but without any $\Delta$-weakly mixing subsets.