math.GT daily digest: 5 new submissions for 27 July 2026

New submissions

Local -semi-rigidity of meandering-hyperbolic actions

• Authors: Sungwoon Kim
• arXiv: 2607.22216
• Subjects: Geometric Topology (math.GT); Group Theory (math.GR)

Meandering hyperbolicity, introduced by Kapovich, Kim, and Lee, extends classical hyperbolicity beyond the setting of word-hyperbolic groups. In this paper, we prove that every meandering-hyperbolic action is locally semi-rigid in the topology. This extends the previously established stability theory in the Lipschitz topology to the full topology. Consequently, we recover the -local semi-rigidity of both boundary actions of word-hyperbolic groups and cocompact lattices in semisimple Lie groups from a single dynamical principle.

Non-existence of Cannon-Thurston maps for hierarchically hyperbolic groups

• Authors: Funda Gültepe; Ravi Tomar
• arXiv: 2607.22447
• Subjects: Geometric Topology (math.GT); Group Theory (math.GR); Metric Geometry (math.MG)
• Comments: 18 pages. Comments are welcome!

We prove that Cannon--Thurston maps do not exist for hyperbolic normal subgroups of hierarchically hyperbolic groups, both for Morse and hierarchically hyperbolic boundaries. This result includes a large class of normal subgroups of the mapping class group of a surface of genus . As a corollary, we also prove a non-existence result for subgroups of most free by cyclic groups. Moreover, we prove that the visual boundary of a CAT(0) group with isolated flats is homeomorphic to the relatively hierarchically hyperbolic boundary of the space it acts on, thus recovering a previously known non-existence result for CAT(0) groups with isolated flats.

Cross submissions

Distortion in the group of locally monotone homeomorphisms of a Cantor set and in the group of generalized interval exchange transformations

• Authors: Nancy Guelman; Emmanuel Militon
• arXiv: 2607.22066
• Cross-list from: math.DS
• Subjects: Dynamical Systems (math.DS); Group Theory (math.GR); Geometric Topology (math.GT)

Let f be either a generalized interval exchange transformation or a locally monotone homeomorphism of a Cantor subset of the real line. In this article, we prove that the following are equivalent. 1. The number of discontinuities of f^n is bounded. 2. There exists n such that the element f is conjugate to the restriction to a closed invariant subset of a disjoint union of n circles of a homeomorphism of this disjoint union of circles. 3. The element f is distorted in the group of generalized interval exchange transformations or in the group of locally monotone homeomorphisms of the Cantor subset.

Nuancing the unicity of -rationals

• Authors: Perrine Jouteur; Olga Paris-Romaskevich; Alexander Thomas
• arXiv: 2607.22308
• Cross-list from: math.QA
• Subjects: Quantum Algebra (math.QA); Combinatorics (math.CO); Geometric Topology (math.GT)
• Comments: 14 pages

We prove unicity of -rational numbers up to conjugacy, using character varieties. Despite the unicity, we exhibit a two-parameter family of deformations of rationals with a modular symmetry. We prove that there are exactly two deformations which deliver the usual -integers: the original -rationals defined by Morier-Genoud and Ovsienko, and another new one. Although the new family can be obtained by conjugacy from the old one, new positivity properties appear. In addition, this new family provides a direct computation of the Jones polynomial of rational knots.

A counterexample for the polar conjecture of Spencer-Brown

• Authors: Scott Baldridge; Louis H. Kauffman; Ben McCarty
• arXiv: 2607.22398
• Cross-list from: math.CO
• Subjects: Combinatorics (math.CO); Geometric Topology (math.GT)
• Comments: 14 pages, multiple figures

In 1976, George Spencer-Brown announced a proof of the four color theorem, using operations on Tait colorings for trivalent plane graphs. In subsequent work he formulated these operations in terms of an algorithm that he called a parity-pass and claimed that when the parity pass algorithm is performed on a non-polar pentagon region, it necessarily terminates in an edge coloring that is extendable to the entire graph. We provide here a counterexample to show that this claim is false. We then raise questions related to the existence of this sort of counterexample.