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Automorphisms of the Rado meet-tree

Itay Kaplan, Binyamin Riahi, Arturo Rodriguez Fanlo

TL;DR

The paper proves that the automorphism group of the generic meet-tree expansion of an infinite non-unary free Fraïssé limit over a finite relational language is simple; in particular Aut($\mathbb{T}^{\mathrm{R}}$) is simple. The approach adapts Li’s method via back-and-forth arguments that either fix a branch or are fans, combined with cone- and semibranch-independence and a generic mix construction to obtain conormal automorphisms. It develops the notions of cone-independence, semibranch-independence, and generic meet-tree expansions, showing that the mixed age class yields the Fraïssé limit $\mathbb{T}^M$ with branched/pointed variants. The results illuminate the structure of automorphism groups of universal dense meet-trees and raise questions about removing the freeness hypothesis and about the generically ordered case.

Abstract

We prove that the group of automorphisms of the generic meet-tree expansion of an infinite non-unary free Fraïssé limit over a finite relational language is simple. As a prototypical case, the group of automorphism of the Rado meet-tree (i.e. the Fraïssé limit of finite graphs which are also meet-trees) is simple.

Automorphisms of the Rado meet-tree

TL;DR

The paper proves that the automorphism group of the generic meet-tree expansion of an infinite non-unary free Fraïssé limit over a finite relational language is simple; in particular Aut() is simple. The approach adapts Li’s method via back-and-forth arguments that either fix a branch or are fans, combined with cone- and semibranch-independence and a generic mix construction to obtain conormal automorphisms. It develops the notions of cone-independence, semibranch-independence, and generic meet-tree expansions, showing that the mixed age class yields the Fraïssé limit with branched/pointed variants. The results illuminate the structure of automorphism groups of universal dense meet-trees and raise questions about removing the freeness hypothesis and about the generically ordered case.

Abstract

We prove that the group of automorphisms of the generic meet-tree expansion of an infinite non-unary free Fraïssé limit over a finite relational language is simple. As a prototypical case, the group of automorphism of the Rado meet-tree (i.e. the Fraïssé limit of finite graphs which are also meet-trees) is simple.
Paper Structure (5 sections, 49 theorems, 2 equations, 2 figures)

This paper contains 5 sections, 49 theorems, 2 equations, 2 figures.

Table of Contents

  1. Preliminaries on meet-trees
  2. Cone-independence
  3. Generic meet-tree expansions
  4. Back-and-forth arguments
  5. Main results

Key Result

Lemma 1.1

Let $T$ be a meet-tree and $\Gamma\subseteq T$. Then, $\Gamma$ is a semibranch if and only if $\Gamma$ is a branch or $\Gamma=T_{\leq \gamma}$ where $\gamma=\max\Gamma$ and $\pi_\Gamma(a)=a\mathbin{\text{$\wedge$}} \gamma$ for all $a\in T$.

Figures (2)

  • Figure 1: Four points Lemma.
  • Figure 2: Example of $\bar{a}\mathop{{ \hbox{$\mid$$\smile$$\gamma$$c$} }}\bar{b}$.

Theorems & Definitions (117)

  • Lemma 1.1
  • Corollary 1.2
  • Lemma 1.3
  • Lemma 1.4
  • proof
  • Corollary 1.5
  • Corollary 1.6
  • proof
  • Corollary 1.7
  • proof
  • ...and 107 more