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Orbit problems for free-abelian times free groups and related families

André Carvalho, Jordi Delgado

Abstract

We prove that the Brinkmann Problems (BrP & BrCP) and the twisted-conjugacy Problem (TCP) are decidable for any endomorphism of a free-abelian times free (FATF) group Fn x Z^m. Furthermore, we prove the decidability of the two-sided Brinkmann conjugacy problem (2BrCP) for monomorphisms of FATF groups (and combine it with TCP) to derive the decidability of the conjugacy problem for ascending HNN extensions of FATF groups.

Orbit problems for free-abelian times free groups and related families

Abstract

We prove that the Brinkmann Problems (BrP & BrCP) and the twisted-conjugacy Problem (TCP) are decidable for any endomorphism of a free-abelian times free (FATF) group Fn x Z^m. Furthermore, we prove the decidability of the two-sided Brinkmann conjugacy problem (2BrCP) for monomorphisms of FATF groups (and combine it with TCP) to derive the decidability of the conjugacy problem for ascending HNN extensions of FATF groups.
Paper Structure (5 sections, 19 theorems, 34 equations, 1 table)

This paper contains 5 sections, 19 theorems, 34 equations, 1 table.

Table of Contents

  1. Free-abelian times free groups and their endomorphisms
  2. The Brinkmann problem
  3. The conjugacy problem for ascending HNN extensions
  4. Twisted conjugacy
  5. Two-sided Brinkmann & the conjugacy problem

Key Result

Proposition 1.1

Let $m\geq 1,n\geq 2$. The following is a complete list of the endomorphisms of ${\mathbb{G}} = \mathbb{F} \times \mathbb{Z}^m$:

Theorems & Definitions (34)

  • Proposition 1.1
  • Corollary 1.2
  • Corollary 1.3
  • Proposition 1.4
  • Remark 1.5
  • Proposition 1.6
  • proof
  • Proposition 1.7
  • Remark 2.1
  • Definition 2.2
  • ...and 24 more