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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
##### Abstract

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 $\mathbb{𝕂}\left[{t}^{±1}\right]$ for $\mathbb{𝕂}$ 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.

##### Keywords
Artin groups, homology
##### Mathematical Subject Classification
Primary: 20F36, 20F65, 20J05, 57M07, 57M10
Secondary: 05C69