# Residual finiteness and strict distortion of cyclic subgroups of solvable groups

## TL;DR

This paper establishes polynomial lower bounds on residual finiteness for certain solvable groups with exponentially distorted cyclic subgroups, improving upon previous results and providing new bounds for groups with infinite Prüfer rank.

## Contribution

It introduces new polynomial lower bounds for residual finiteness in solvable groups with exponentially distorted cyclic subgroups, including polycyclic and Baumslag-Solitar groups.

## Key findings

- Polynomial lower bounds for residual finiteness in specific solvable groups.
- Improved bounds over previous literature for polycyclic and Baumslag-Solitar groups.
- First nontrivial bounds for groups with infinite Prüfer rank under certain conditions.

## Abstract

We provide polynomial lower bounds for residual finiteness of residually finite, finitely generated solvable groups that admit infinite order elements in the Fitting subgroup of strict distortion at least exponential. For this class of solvable groups which include polycyclic groups with a nontrivial exponential radical and the metabelian Baumslag-Solitar groups, we improve the lower bounds found in the literature. Additionally, for the class of residually finite, finitely generated solvable groups of infinite Pr\"{u}fer rank that satisfy the conditions of our theorem, we provide the first nontrivial lower bounds.