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Direct and Inverse Problems in Baumslag-Solitar Group $BS(1,3)$

Sandeep Singh, Ramandeep Kaur

Abstract

For integers $m$ and $n$, the Baumslag-Solitar groups, denoted as $BS(m,n)$, are groups generated by two elements with a single defining relation: $BS(m,n) = \langle a, b | a^mb=ba^n\rangle$. The sum of dilates, denoted as $r \cdot A + s \cdot B$ for integers $r$ and $s$, is defined as $\{ra + sb; a\in A, b\in B\}$. In 2014, Freiman et al. \cite{freiman} derived direct and inverse results for sums of dilates and applied these findings to address specific direct and inverse problems within Baumslag-Solitar groups, assuming suitable small doubling properties. In 2015, Freiman et al. \cite{freiman15} tackled the general problem of small doubling types in a monoid, a subset of the Baumslag-Solitar group $BS(1,2)$. This paper extends these investigations to solve the analogous problem for the Baumslag-Solitar group $BS(1,3)$.

Direct and Inverse Problems in Baumslag-Solitar Group $BS(1,3)$

Abstract

For integers and , the Baumslag-Solitar groups, denoted as , are groups generated by two elements with a single defining relation: . The sum of dilates, denoted as for integers and , is defined as . In 2014, Freiman et al. \cite{freiman} derived direct and inverse results for sums of dilates and applied these findings to address specific direct and inverse problems within Baumslag-Solitar groups, assuming suitable small doubling properties. In 2015, Freiman et al. \cite{freiman15} tackled the general problem of small doubling types in a monoid, a subset of the Baumslag-Solitar group . This paper extends these investigations to solve the analogous problem for the Baumslag-Solitar group .
Paper Structure (3 sections, 12 theorems, 35 equations)

This paper contains 3 sections, 12 theorems, 35 equations.

Table of Contents

  1. Introduction
  2. Some Important Lemmas
  3. Proof of Main Theorem

Key Result

Theorem 1.1

If $S$ is a finite non-abelian subset of $BS^{+}(1,2)$ of size $|S|=k$, then the following statements hold: (a) The size of $S^2$ satisfies $|S^2|\geq 3k-2.$ (b) If $|S^2|=(3k-2)+h<\frac{7}{2}k-4,$ then there exists a finite set of integers $A\subseteq \mathbb{Z}$ such that $S=ba^A$ and the set $A$

Theorems & Definitions (17)

  • Theorem 1.1: freiman15
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4: LS,S
  • Theorem 1.5: freiman
  • Corollary 1.1: freiman
  • Theorem 1.6: freiman15
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • ...and 7 more