OpenFrontiers
Join
← Back to the index
OP-11821Status unassessedLogic and foundations

Outer automorphisms of oligomorphic groups — Problem 2.1

Outer automorphisms of oligomorphic groups — Problem 2.1. Current research status is unassessed. The dated source poses this question; inclusion does not certify that it remains open today.

MSC 03 · Mathematical logic and foundations

Editorial MSC2020 mapping of Outer automorphisms of oligomorphic groups — Problem 2.1. https://arxiv.org/html/2608.26628v2#S2.Thmproblem1 Not an author-supplied code.

Licensed source statementAs of 2608.26628v2; current status unassessedSource & evidence ↓
Read original source ↗ 0Submit progressLean 4 ↓Discussion (0)

Problem statement

Problem 2.1.

Let MM be an ω\omega-categorical structure in a first order finite language. Is the group Out⁡(G)\operatorname{Out}(G) finite?

Lean 4

Not yet attached

No Lean statement or proof is attached yet. Have a formalization? Share its source to add it to this problem.

Share a formalization →Verification guide ↗

Background

Oligomorphic groups

Let MM be a countably infinite ω\omega-categorical structure in a finite first-order signature, and let

G=Aut⁡(M),G=\operatorname{Aut}(M),

equipped with the topology of pointwise convergence. Let Aut⁡(G)\operatorname{Aut}(G) denote the group of continuous automorphisms of the topological group GG, and let

Out⁡(G):=Aut⁡(G)/Inn⁡(G)\operatorname{Out}(G):=\operatorname{Aut}(G)/\operatorname{Inn}(G)

be its outer automorphism group.

For background see [2], where it is shown that this group is countable. For some classical examples like M=(Q,<)M=(\mathbb{Q},<) [3], the statement of the problem is true; in this case Out(G){\rm Out}(G) has two elements, given by the identity and conjugation by x↦−xx\mapsto-x.

Progress & source assessment

Current research status is unassessed. The dated source poses this question; inclusion does not certify that it remains open today.

References
  1. [1]Barmpalias, George; Gao, Su; He, Jialiang; Kihara, Takayuki; Nies, Andre; Slaman, Theodore; Tran, Chieu-Minh; Turetsky, Daniel; Welch, Philip; Yu, Liang; Zhang, Hang. Tianyuan Workshop — Open Problems in Mathematical Logic (2026).; Outer automorphisms of oligomorphic groups — Problem 2.1 linkhttp://creativecommons.org/licenses/by/4.0/; attributed adaptation; HTML converted to Markdown with original mathematical notation and source links.

Recorded progress

Add yours →

Nothing recorded yet. A pointer to a paper the entry is missing counts.

Machine attempts

0 recorded

Attempts run against the statement below and are recorded in full, including the failures. The point is to stop the next model — and the next person — from re-walking a road that is already known to end. Each attempt costs 25 credits.

No machine attempts recorded.

Sign in to run an attempt. New accounts start with enough credits for ten.

Discussion

0 notes

Notes are for clarifying the statement, flagging a missing reference, or recording why an approach fails. Claimed solutions belong in a submission, where they can be reviewed.

No notes yet.

Sign in to add a note.