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On the almost palindromic width of certain free constructions of groups

Krishnendu Gongopadhyay, Shrinit Singh

Abstract

We provide a general structural criterion implying that a group has infinite $m$-almost palindromic width. In particular, we prove that both HNN extensions and free products exhibit infinite $m$-almost palindromic width, with the unique exception of the infinite dihedral group among free products. This framework extends and strengthens the results of \cite{MS} and \cite{GK}.

On the almost palindromic width of certain free constructions of groups

Abstract

We provide a general structural criterion implying that a group has infinite -almost palindromic width. In particular, we prove that both HNN extensions and free products exhibit infinite -almost palindromic width, with the unique exception of the infinite dihedral group among free products. This framework extends and strengthens the results of \cite{MS} and \cite{GK}.

Paper Structure

This paper contains 8 sections, 20 theorems, 37 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['main']}
  3. Proof of Theorem \ref{['main']}
  4. Proof of Theorem \ref{['HNN']}
  5. Proof of Theorem \ref{['HNN']}
  6. Proof of Theorem \ref{['freeprod']}
  7. Proof of Theorem \ref{['freeprod']}
  8. Almost Palindromic Width for Certain Amalgamated free product

Key Result

Theorem 1.1

Let $G = \langle A \rangle$. If $G$ admits an unbounded quasimorphism $f$ that is bounded above on palindromes over $A$, then $\mathrm{pw}_m(G, A)$ is infinite.

Theorems & Definitions (29)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Corollary 1.5
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Lemma 3.1
  • Definition 3.2
  • ...and 19 more