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JSJ decompositions and polytopes for two-generator one-relator groups

Giles Gardam, Dawid Kielak, Alan D. Logan

TL;DR

This work links the $\mathcal{Z}_{\max}$-JSJ decomposition of BS-free two-generator one-relator groups with the Friedl--Tillmann polytope, showing that deformations and splittings can be read off from relators and their automorphic orbits. It develops a robust framework for algebraically hyperbolic (CSA$_{\mathbb{Q}}$) groups, proving existence and uniqueness of $\mathcal{Z}_{\max}$-JSJ trees in the relevant settings and establishing a canonical, rigid-vertex, loop-edge structure in the non-trivial case. The authors then prove a main theorem equating non-trivial $\mathcal{Z}_{\max}$-JSJ decompositions with the presence of a short automorphic image of the relator in a distinguished subgroup, and with the Friedl--Tillmann polytope being a straight line (not a point); torsion cases are treated analogously. Consequences include quadratic-time algorithms to compute and detect $\mathcal{Z}_{\max}$-JSJ decompositions, clear descriptions of outer automorphism groups, and a precise understanding of how these invariants behave under powers in the relator, providing practical tools for isomorphism and automorphism problems in this class of groups.

Abstract

We provide a direct connection between the Z_{max} (or essential) JSJ decomposition and the Friedl--Tillmann polytope of a hyperbolic two-generator one-relator group with abelianisation of rank $2$. We deduce various structural and algorithmic properties, like the existence of a quadratic-time algorithm computing the Z_{max}-JSJ decomposition of such groups.

JSJ decompositions and polytopes for two-generator one-relator groups

TL;DR

This work links the -JSJ decomposition of BS-free two-generator one-relator groups with the Friedl--Tillmann polytope, showing that deformations and splittings can be read off from relators and their automorphic orbits. It develops a robust framework for algebraically hyperbolic (CSA) groups, proving existence and uniqueness of -JSJ trees in the relevant settings and establishing a canonical, rigid-vertex, loop-edge structure in the non-trivial case. The authors then prove a main theorem equating non-trivial -JSJ decompositions with the presence of a short automorphic image of the relator in a distinguished subgroup, and with the Friedl--Tillmann polytope being a straight line (not a point); torsion cases are treated analogously. Consequences include quadratic-time algorithms to compute and detect -JSJ decompositions, clear descriptions of outer automorphism groups, and a precise understanding of how these invariants behave under powers in the relator, providing practical tools for isomorphism and automorphism problems in this class of groups.

Abstract

We provide a direct connection between the Z_{max} (or essential) JSJ decomposition and the Friedl--Tillmann polytope of a hyperbolic two-generator one-relator group with abelianisation of rank . We deduce various structural and algorithmic properties, like the existence of a quadratic-time algorithm computing the Z_{max}-JSJ decomposition of such groups.

Paper Structure

This paper contains 10 sections, 31 theorems, 22 equations, 1 figure.

Table of Contents

  1. Introduction
  2. JSJ-theory and algebraically hyperbolic groups
  3. Algebraically hyperbolic groups
  4. Existence
  5. Uniqueness of deformation space
  6. Two-generated groups
  7. One-relator groups with torsion
  8. Splittings and Friedl--Tillmann polytopes
  9. The main theorem
  10. Consequences of the main theorem

Key Result

Theorem 1

Let $G$ be a $\operatorname{BS}$-free group admitting a two-generator one-relator presentation $\mathcal{P}=\langle a, b\mid R\rangle$ with $R\in F(a, b)'\setminus\{1\}$. The following are equivalent.

Figures (1)

  • Figure 1: To obtain the Friedl--Tillmann polytope, trace the reduced word $(a^2b^2a^{-1}b^{-1}a^{-1}b^{-1})^n$ on the $ab$-plane to obtain a closed loop $\gamma$, as in the first diagram (this is independent of $n$). Take the convex hull of $\gamma$, as in the second diagram; this is a polytope $P'$. Then take the bottom-left corner of all squares contained in $\gamma$ that touch the vertices of $P'$, as in the third diagram. The Friedl--Tillmann polytope $P$ is the polytope with these points as vertices, as in the fourth diagram. Note that the Friedl--Tillmann polytope is in fact a "marked" polytope, but we only care about the shape so we have omitted these details from this example. In the third diagram we took the bottom-left corner of the squares; this is different from Friedl and Tillmann who take the centre points of these squares, but this is not an issue because the polytope is only well-defined up to translation.

Theorems & Definitions (56)

  • Theorem 1: Theorem \ref{['thm:flexibilityHyperbolicBODYVERSION']}
  • Example 1.1
  • Theorem 2: Theorem \ref{['thm:flexibilityWithTorsionBODYVERSION']}
  • Corollary 3: Corollary \ref{['corol:JSJformBODYVERSION']}
  • Example 1.2
  • Corollary 4: Corollary \ref{['corol:JSJcompBODYVERSION']}
  • Corollary 5: Corollary \ref{['corol:JSJdetectBODYVERSION']}
  • Corollary 6: Corollary \ref{['corol:OutdetectBODYVERSION']}
  • Corollary 7: Corollary \ref{['corol:OutCommensurabilityBODYVERSION']}
  • Lemma 2.1
  • ...and 46 more