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Uniform first order interpretation of the second order theory of countable groups of homeomorphisms

Thomas Koberda, J. de la Nuez González

TL;DR

This work establishes that the FO theory of the homeomorphism group of a compact manifold $M$ uniformly interprets the full second-order theory of countable groups of homeomorphisms, with a dimension-bounded uniform construction. Central to the method is the conservative expansion by HS sorts, beginning with $ ext{HS}_0(M)\cong ext{Homeo}(M)$ and recursively defining higher sorts as sequences, enabling parameter-free, uniform interpretations of countable sequences, points, regular open sets, and eventually topological constructs. The authors derive broad group-theoretic consequences (e.g., interpretation of $ ext{Mod}(M)$ and intermediate subgroups, finite generation, linearity, amenability, and property (T)) and recover descriptive set-theoretic structure (open/closed sets, Borel and projective hierarchies) inside $ ext{Homeo}(M)$; they also prove Rice-type theorems and independence results showing undefinability in second-order arithmetic and potential independence from ZFC. Altogether, the paper provides a robust bridge from first-order logic on homeomorphism groups to rich second-order arithmetic, descriptive set theory, and computability-theoretic phenomena in group theory and geometry across manifolds of bounded dimension.

Abstract

We show that the first order theory of the homeomorphism group of a compact manifold interprets the full second order theory of countable groups of homeomorphisms of the manifold. The interpretation is uniform across manifolds of bounded dimension. As a consequence, many classical problems in group theory and geometry (e.g.~the linearity of mapping classes of compact $2$--manifolds) are encoded as elementary properties of homeomorphism groups of manifolds. Furthermore, the homeomorphism group uniformly interprets the Borel and projective hierarchies of the homeomorphism group, which gives a characterization of definable subsets of the homeomorphism group. Finally, we prove analogues of Rice's Theorem from computability theory for homeomorphism groups of manifolds. As a consequence, it follows that the collection of sentences that isolate the homeomorphism group of a particular manifold, or that isolate the homeomorphism groups of manifolds in general, is not definable in second order arithmetic, and that membership of particular sentences in these collections cannot be proved in ZFC.

Uniform first order interpretation of the second order theory of countable groups of homeomorphisms

TL;DR

This work establishes that the FO theory of the homeomorphism group of a compact manifold uniformly interprets the full second-order theory of countable groups of homeomorphisms, with a dimension-bounded uniform construction. Central to the method is the conservative expansion by HS sorts, beginning with and recursively defining higher sorts as sequences, enabling parameter-free, uniform interpretations of countable sequences, points, regular open sets, and eventually topological constructs. The authors derive broad group-theoretic consequences (e.g., interpretation of and intermediate subgroups, finite generation, linearity, amenability, and property (T)) and recover descriptive set-theoretic structure (open/closed sets, Borel and projective hierarchies) inside ; they also prove Rice-type theorems and independence results showing undefinability in second-order arithmetic and potential independence from ZFC. Altogether, the paper provides a robust bridge from first-order logic on homeomorphism groups to rich second-order arithmetic, descriptive set theory, and computability-theoretic phenomena in group theory and geometry across manifolds of bounded dimension.

Abstract

We show that the first order theory of the homeomorphism group of a compact manifold interprets the full second order theory of countable groups of homeomorphisms of the manifold. The interpretation is uniform across manifolds of bounded dimension. As a consequence, many classical problems in group theory and geometry (e.g.~the linearity of mapping classes of compact --manifolds) are encoded as elementary properties of homeomorphism groups of manifolds. Furthermore, the homeomorphism group uniformly interprets the Borel and projective hierarchies of the homeomorphism group, which gives a characterization of definable subsets of the homeomorphism group. Finally, we prove analogues of Rice's Theorem from computability theory for homeomorphism groups of manifolds. As a consequence, it follows that the collection of sentences that isolate the homeomorphism group of a particular manifold, or that isolate the homeomorphism groups of manifolds in general, is not definable in second order arithmetic, and that membership of particular sentences in these collections cannot be proved in ZFC.
Paper Structure (21 sections, 27 theorems, 23 equations, 2 figures)

This paper contains 21 sections, 27 theorems, 23 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. Group theoretic consequences of the main results
  4. Descriptive set theory
  5. Undefinability and independence
  6. Organization of the paper
  7. Background
  8. Results from geometric topology of manifolds
  9. Results about the first order theory of homeomorphism groups of manifolds
  10. Hereditarily sequential sets of homeomorphisms of a manifold
  11. Interpreting $\mathrm{HS}_0(M)$
  12. Interpreting $\mathcal{H}$ within $\mathop{\mathrm{HS}}\nolimits_0(M)$
  13. Interpreting the sorts $\mathop{\mathrm{HS}}\nolimits_n(M)$ for $n\geq 1$
  14. Predicates for manipulating $\mathop{\mathrm{HS}}\nolimits(M)$
  15. Intermediate subgroups, mapping class groups, and Theorem \ref{['thm:gp-thy-conseq']}
  16. ...and 6 more sections

Key Result

Theorem 1.1

Let $D\geq 1$ be a natural number, and let Then there is a conservative expansion of the language of group theory and a uniform interpretation of the union of the sorts $\mathop{\mathrm{HS}}\nolimits(M)$ in $\mathcal{H}$ that is valid for all manifolds $M$ with $\dim M\leq D$. The elements in the sort $\mathop{\mathrm{HS}}\nolimits_0(M)$ can

Figures (2)

  • Figure 1: A schematic of the scratchpad; here we draw the image of one chart $U$ in the atlas (which need not actually be a disk) in $W$, and the image under $g_0$. The iterates under $g_0$ continue to infinity.
  • Figure 2: A schematic of two components in $U^i$. The sets $P$ meets $U^i$ and isolates a unique point in it.

Theorems & Definitions (42)

  • Theorem 1.1
  • Proposition 1.2
  • Proposition 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Proposition 2.1: See Chapter 3 in coornaert2015, Theorem IV.2 in HW1941, Theorem 3 in ostrand71, Section 6.1 in dlNKK22
  • proof : Proof of Proposition \ref{['prop:bounded cover']}
  • Corollary 2.2
  • ...and 32 more