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BNSR-Invariants of Surface Houghton Groups

Noah Torgerson, Jeremy West

TL;DR

The paper addresses the computation of the BNSR-invariants for the pure surface Houghton groups, situating them in the framework of asymptotically rigid mapping class groups. It builds a CAT(0) Stein--Farley cube complex and applies a Zaremsky-style Morse analysis to obtain a precise description: for a nonzero character $\chi$ in ascending standard form with $m(\chi)$ nonzero coefficients, $[\chi]$ lies in $Σ^{m(χ)-1}$ but not in $Σ^{m(χ)}$. This yields a sharp finiteness-length criterion: subgroups with maximal finiteness length that intersect the commutator subgroup in finite index are finite index in the ambient group. The work also analyzes co-Hopfian and dis-co-Hopfian phenomena, including braided Houghton groups and open questions for the pure surface case, connecting these invariants to a broader family of asymptotically rigid mapping class groups.

Abstract

The surface Houghton groups $\mathcal{H}_{n}$ are a family of groups generalizing Houghton groups $H_n$, which are constructed as asymptotically rigid mapping class groups. We give a complete computation of the BNSR-invariants $Σ^{m}(P\mathcal{H}_{n})$ of their intersection with the pure mapping class group. To do so, we prove that the associated Stein--Farley cube complex is CAT(0), and we adapt Zaremsky's method for computing the BNSR-invariants of the Houghton groups. As a consequence, we give a criterion for when subgroups of $H_n$ and $P\mathcal{H}_{n}$ having the same finiteness length as their parent group are finite index. We also discuss the failure of some of these groups to be co-Hopfian.

BNSR-Invariants of Surface Houghton Groups

TL;DR

The paper addresses the computation of the BNSR-invariants for the pure surface Houghton groups, situating them in the framework of asymptotically rigid mapping class groups. It builds a CAT(0) Stein--Farley cube complex and applies a Zaremsky-style Morse analysis to obtain a precise description: for a nonzero character in ascending standard form with nonzero coefficients, lies in but not in . This yields a sharp finiteness-length criterion: subgroups with maximal finiteness length that intersect the commutator subgroup in finite index are finite index in the ambient group. The work also analyzes co-Hopfian and dis-co-Hopfian phenomena, including braided Houghton groups and open questions for the pure surface case, connecting these invariants to a broader family of asymptotically rigid mapping class groups.

Abstract

The surface Houghton groups are a family of groups generalizing Houghton groups , which are constructed as asymptotically rigid mapping class groups. We give a complete computation of the BNSR-invariants of their intersection with the pure mapping class group. To do so, we prove that the associated Stein--Farley cube complex is CAT(0), and we adapt Zaremsky's method for computing the BNSR-invariants of the Houghton groups. As a consequence, we give a criterion for when subgroups of and having the same finiteness length as their parent group are finite index. We also discuss the failure of some of these groups to be co-Hopfian.
Paper Structure (10 sections, 36 theorems, 8 equations, 9 figures)

This paper contains 10 sections, 36 theorems, 8 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Definitions
  3. The (Pure) Surface Houghton Group
  4. The Contractible Cube Complex
  5. Characters and $\Sigma$-Invariants
  6. The Stein--Farley complex $X_n$ is CAT(0)
  7. $\Sigma$-invariants of $P\mathcal{H}\xspace_{n}$
  8. Inclusion
  9. Exclusion
  10. Subgroups of maximal finiteness length

Key Result

Theorem 4.1

Let $\chi$ be a non-zero character of $P\mathcal{H}\xspace_{n}$. Then $[\chi]\in \Sigma^{m(\chi)-1}(P\mathcal{H}\xspace_{n})\setminus \Sigma^{m(\chi)}(P\mathcal{H}\xspace_{n})$.

Figures (9)

  • Figure 1: The handle shift $\rho_i$ (which pushes from end $i$ to end $n$) being applied to various curves in ends $i$, $n$, and $j$.
  • Figure 2: A collapsed square
  • Figure 3: A 3-wheel with common vertex maximal.
  • Figure 4: Left: the defining surface for $\mathop{\mathrm{Br}}\nolimits H_n$; Right: the surface $\Sigma'$
  • Figure 5: The central curve $\beta$ in a piece which is fixed by the image of $\mathop{\mathrm{Br}}\nolimits H_{n}$ in $P\mathcal{H}\xspace_{n}$
  • ...and 4 more figures

Theorems & Definitions (65)

  • Theorem 4.1
  • Theorem 3.9
  • Theorem 5.6
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Remark 2.3
  • Theorem 2.4
  • Corollary 2.5
  • Lemma 2.6
  • ...and 55 more