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Computable Scott sentences and the weak Whitehead problem for finitely presented groups

Gianluca Paolini

Abstract

We prove that if $A$ is a computable Hopfian finitely presented structure, then $A$ has a computable $d$-$Σ_2$ Scott sentence if and only if the weak Whitehead problem for $A$ is decidable. We use this to infer that every hyperbolic group as well as any polycyclic-by-finite group has a computable $d$-$Σ_2$ Scott sentence, thus covering two main classes of finitely presented groups. Our proof also implies that every weakly Hopfian finitely presented group is strongly defined by its $\exists^+$-types, a question which arose in a different context.

Computable Scott sentences and the weak Whitehead problem for finitely presented groups

Abstract

We prove that if is a computable Hopfian finitely presented structure, then has a computable - Scott sentence if and only if the weak Whitehead problem for is decidable. We use this to infer that every hyperbolic group as well as any polycyclic-by-finite group has a computable - Scott sentence, thus covering two main classes of finitely presented groups. Our proof also implies that every weakly Hopfian finitely presented group is strongly defined by its -types, a question which arose in a different context.
Paper Structure (3 sections, 12 theorems, 10 equations)

This paper contains 3 sections, 12 theorems, 10 equations.

Table of Contents

  1. Introduction
  2. The proof of the characterization
  3. Groups defined by their types

Key Result

Theorem 1.4

Let $A$ be a computable Hopfian f.p. structure. Then $A$ has a computable $d$-$\Sigma_2$ Scott sentence iff the weak Whitehead problem for $A$ is decidable.

Theorems & Definitions (34)

  • Definition 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Corollary 1.6
  • Conjecture 1.7
  • Theorem 1.8
  • Remark 2.2
  • Definition 2.3
  • Lemma 2.4
  • proof
  • ...and 24 more