Weakly G-slim complexes and the non-positive immersion property for generalized Wirtinger presentations

2025/06/23 by Agustín Nicolás Barreto, Barreto, Agustín Nicolás, Elías Gabriel Minian +1
Computer Science · Mathematics · #20F05 #20F65 #57M05 #57M07 #57M10 #Advanced Operator Algebra Research #FOS: Mathematics #Geometric Topology (math.GT) #Group Theory (math.GR) #Homotopy and Cohomology in Algebraic Topology #Topological and Geometric Data Analysis

paper · pdf · doi:10.48550/arxiv.2506.19105

openalex publication_date 2025/06/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We investigate weakly G-slim complexes, a more flexible variant of Helfer and Wise's slim complexes, which can be defined on any regular G-covering. We prove that if a 2-complex X associated to a group presentation is weakly H-slim for some regular H-covering, and π₁X is left-orderable, then X is slim and, in particular, it has non-positive immersions. We apply this result to study conditions on generalized Wirtinger presentations that guarantee the non-positive immersion property for their associated 2-complexes. This class of presentations includes classical Wirtinger presentations of (perhaps high-dimensional) knots in spheres, some classes of Adian presentations and LOTs presentations. On the one hand, our results provide a wide variety of examples of presentation complexes with the non-positive immersion property via a condition that can be checked with a simple algorithm. On the other hand, as an application of these methods, we derive an extension of a result by Wise on Adian presentations and prove a stronger formulation of a result of Howie on LOT presentations.

Citations

• Local indicability of groups with homology circle presentations
• A 2-complex with contracting non-positive immersions and positive maximal irreducible curvature
• On the coherence of one-relator groups and their group algebras
• Rational curvature invariants for 2-complexes
• Uniform negative immersions and the coherence of one-relator groups
• Coherence and one-relator products of locally indicable groups
• One-relator groups with torsion are coherent
• Ordered groups and topology
• Stackings and the W-cycle conjecture
• Counting cycles in labeled graphs: The nonpositive immersion property\n for one-relator groups

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• Most unexposed taut one-relator presentation 2-complexes are finitely\n unsplittable
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