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Module structure of the homology of right-angled Artin kernels

Enrique Artal Bartolo, José Ignacio Cogolludo-Agustín, Santiago López de Medrano and Daniel Matei

Algebraic & Geometric Topology 22 (2022) 2775–2803
DOI: 10.2140/agt.2022.22.2775

We study the module structure of the homology of Artin kernels, ie kernels of nonresonant characters from right-angled Artin groups onto the integer numbers, where the module structure is with respect to the ring 𝕂[t±1] for 𝕂 a field of characteristic zero. Papadima and Suciu determined some part of this structure by means of the flag complex of the graph of the Artin group. We provide more properties of the torsion part of this module, eg the dimension of each primary part and the maximal size of Jordan forms (if we interpret the torsion structure in terms of a linear map). These properties are stated in terms of homology properties of suitable filtrations of the flag complex and suitable double covers of an associated toric complex.

Artin groups, homology
Mathematical Subject Classification
Primary: 20F36, 20F65, 20J05, 57M07, 57M10
Secondary: 05C69
Received: 27 October 2020
Revised: 29 March 2021
Accepted: 28 July 2021
Published: 13 December 2022
Enrique Artal Bartolo
Departamento de Matemáticas, IUMA, Facultad de Ciencias
Universidad de Zaragoza
José Ignacio Cogolludo-Agustín
Departamento de Matemáticas, IUMA, Facultad de Ciencias
Universidad de Zaragoza
Santiago López de Medrano
Instituto de Matemáticas
Universidad Nacional Autónoma de México
Mexico City
Daniel Matei
Institute of Mathematics of the Romanian Academy