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Quasi-isometric rigidity for the solvable Baumslag-Solitar groups, II

Benson Farb, Lee Mosher

TL;DR

The paper establishes quasi-isometric rigidity for the solvable Baumslag–Solitar groups BS(1,n) by building a geometric model X_n and analyzing its two-boundary structure, which yields a representation of any quasi-isometric group G into Bilip(ℝ) × Bilip(ℚ_n). A key step is a Hinkkanen-type result showing uniform quasisimilarity actions on ℝ are bilipschitz conjugate to affine actions, enabling an affine realization of the G-action on the real line. Boundary dynamics are further studied via a biconvergence framework to prove the affine action is virtually faithful and to constrain the stretch dynamics to be infinite cyclic, leading to a description of Γ = θ(G) as a mapping torus and establishing abstract commensurability with BS(1,n). Consequently, any finitely generated G quasi-isometric to BS(1,n) fits into a short exact sequence with finite kernel and a quotient Γ commensurable to BS(1,n), yielding strong quasi-isometric rigidity for these solvable groups. The work also discusses Sullivan–Tukia-type questions in this nonlattice solvable setting and outlines directions for extending rigidity results to related groups and boundary actions.

Abstract

Let BS(1,n)= < a,b: aba^{-1}=b^n >. We prove that any finitely-generated group quasi-isometric to BS(1,n) is (up to finite groups) isomorphic to BS(1,n). We also show that any uniform group of quasisimilarities of the real line is bilipschitz conjugate to an affine group.

Quasi-isometric rigidity for the solvable Baumslag-Solitar groups, II

TL;DR

The paper establishes quasi-isometric rigidity for the solvable Baumslag–Solitar groups BS(1,n) by building a geometric model X_n and analyzing its two-boundary structure, which yields a representation of any quasi-isometric group G into Bilip(ℝ) × Bilip(ℚ_n). A key step is a Hinkkanen-type result showing uniform quasisimilarity actions on ℝ are bilipschitz conjugate to affine actions, enabling an affine realization of the G-action on the real line. Boundary dynamics are further studied via a biconvergence framework to prove the affine action is virtually faithful and to constrain the stretch dynamics to be infinite cyclic, leading to a description of Γ = θ(G) as a mapping torus and establishing abstract commensurability with BS(1,n). Consequently, any finitely generated G quasi-isometric to BS(1,n) fits into a short exact sequence with finite kernel and a quotient Γ commensurable to BS(1,n), yielding strong quasi-isometric rigidity for these solvable groups. The work also discusses Sullivan–Tukia-type questions in this nonlattice solvable setting and outlines directions for extending rigidity results to related groups and boundary actions.

Abstract

Let BS(1,n)= < a,b: aba^{-1}=b^n >. We prove that any finitely-generated group quasi-isometric to BS(1,n) is (up to finite groups) isomorphic to BS(1,n). We also show that any uniform group of quasisimilarities of the real line is bilipschitz conjugate to an affine group.

Paper Structure

This paper contains 18 sections, 17 theorems, 74 equations, 2 figures.

Table of Contents

  1. Geometry and boundaries for $\mathbf{\mathop{\mathrm{BS}}\nolimits(1,n)}$
  2. Quasi-isometries
  3. The 2-complex $X_n$
  4. The upper and lower boundaries of $X_n$
  5. Metrics on $\partial_\ell X_n$ and $\partial^u X_n$
  6. The groups $\mathop{\mathrm{Aff}}\nolimits({\mathbf Q}_n)$ and $\mathop{\mathrm{Sim}}\nolimits({\mathbf Q}_n)$
  7. Boundary maps induced by a quasi-isometry
  8. Representation into the quasi-isometry group
  9. Quasi-isometries and quasi-actions
  10. Consequences for $\mathbf{BS(1,n)}$
  11. Uniform quasisimilarity actions on ${\mathbf R}$
  12. A Hinkkanen Theorem for uniform quasisimilarity groups
  13. Proof of Theorem \ref{['theorem:qshink']}
  14. A virtually faithful affine action on ${\mathbf R}$
  15. Biconvergence groups
  16. ...and 3 more sections

Key Result

Proposition 1.1

There exist actions of $\mathop{\mathrm{BS}}\nolimits(1,n)$ on $X_n$, ${\mathbf H}^2$, $T_n$, and ${\mathbf R}$, and equivariant maps between these spaces as summarized in the following commutative diagram: \xymatrix{ & X_n \ar[dl]_p \ar[dd]^h \ar[dr]^q \\ {{\mathbf H}^2} \ar[dr]_h & & T_n \ar[dl]^h

Figures (2)

  • Figure 1: The point $\pi(x,y,\zeta)$ is the barycenter of the triangle $\Delta(x,y,+\infty)$ in the hyperbolic plane $H \subset X_n$ correponding to $\zeta$.
  • Figure 2: The point $\kappa(x,\eta,\zeta)$ is the intersection of the three sides of the triangle $\Delta(-\infty,\eta,\zeta)$ in the tree $\tau \subset X_n$ correponding to $x$.

Theorems & Definitions (39)

  • Proposition 1.1
  • proof
  • Remark
  • Proposition 1.2
  • Proposition 2.1: QI rigidity condition
  • proof
  • Proposition 2.2
  • Theorem 3.1: Hinkkanen's Theorem
  • Definition : quasisimilarity
  • Theorem 3.2: Quasisimilarity Hinkkanen's Theorem
  • ...and 29 more