Automorphisms of Dlab groups and conjugation in overgroups
Year: 2026
Type: preprint
Abstract: We answer affirmatively the Dlab-group question recorded as Problem 21.149 in the 2026 Kourovka Notebook, signed by A. V. Zenkov with V. M. Kopytov and N. Ya. Medvedev named parenthetically: a Dlab group has an order automorphism which cannot be induced by conjugation in any larger standard real Dlab group. We use the alternating-slope homeomorphism constructed by Gong, Yang, and Zeng, whose theorem treats six rank-one ambient families. The extension here allows arbitrary target slope subgroups and includes the full extended-real groups and their subgroups fixing a right tail. For the compact-interior group with slopes in 2^Z, the obstruction holds along every abstract group embedding. The proof uses its two biorders to recover first-movement points under an embedding, and perfectness removes the affine germ at minus infinity. We also obtain an analogous automorphism for every nontrivial real source slope group, preserving all its biorders, and a biorderable mapping torus with no abstract embedding in any standard real Dlab group. Source repository: https://github.com/eulogica/dlab-automorphism-obstruction
Author: Lupini, Martino
Institution: University of Bologna
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225161328