arXiv subjects

Marcos Escartín Ferrer

Publications and source records attributed to Marcos Escartín Ferrer.

5 recordsLinked to original sources

arXiv2026

Cohomology rings and $p$-local behavior of even Artin groups

We generalize to certain families of even Artin groups several classical results on right-angled Artin groups. In particular, we compute the cohomology ring, describe the pro-$p$ completion, and determine the $p$-Zassenhaus restricted Lie algebra in the FC case. As a by-product, we prove a rigidity result that implies that if two even Artin groups of FC type are isomorphic, then for every prime $p$, the $p$-parts of their defining graphs are isomorphic.

math.GR↗

arXiv2026

Actions with a strong fundamental domain and Sigma invariants of groups

We show that if a group $G$ acts on a contractible $CW$-complex with a contractible strong fundamental domain and the stabilizers of the action satisfy certain properties, then the homological Sigma invariants of the group are either empty or dense in the character sphere. This was known for several families of groups like right-angled Artin groups and pure symmetric automorphisms of free groups. We also exhibit some applications including the generalization of the previous fact to pure symmetric automorphisms of right-angled Artin groups.

math.GR↗

arXiv2025

On the $\ell^2$-Betti numbers and algebraic fibring of the (outer) automorphism group of a right-angled Artin group

We compute the first $\ell^2$-Betti number of the automorphism and outer automorphism groups of arbitrary right-angled Artin groups (RAAGs), providing a complete characterization of when it is non-zero. We also analyse the algebraic fibring of the pure symmetric automorphism groups $\mathrm{PSA}(A_Γ)$ and $\mathrm{PSO}(A_Γ)$ and the virtual algebraic fibring of $\mathrm{Out}(A_Γ)$ in the case when $A_Γ$ admits no non-inner partial conjugation. In the transvection-free case, we show that $β_1^{(2)}(\mathrm{Out}(A_Γ)) = 0$ if and only if $\mathrm{Out}(A_Γ)$ virtually fibres.

math.GR↗

arXiv2025

On the $Σ^1$ and $Σ^2$-invariants of Artin groups

We prove the $Σ^1$-conjecture for two families of Artin groups: Artin groups such that there exists a prime number $p$ dividing $\frac{l(e)}{2}$ for every edge $e$ with even label $>2$ and balanced Artin groups. The family of balanced Artin groups extends two previously studied families: the one considered by Kochloukova in arXiv:2009.14269, and the family of coherent Artin groups. We state a conjecture on the $Σ^2$-invariant for Artin groups satisfying the $K(π,1)$-conjecture. The conjecture is proven to be true for two significant families: $2$-dimensional and coherent Artin groups. In the $2$-dimensional case we are able to compute $Σ^n$ for all $n\geq 2$ and to derive finiteness properties of the derived subgroup.

math.GR↗

arXiv2023

On the $Σ$-invariants of Artin groups satisfying the $K(π,1)$-conjecture

We consider $Σ$-invariants of Artin groups that satisfy the $K(π,1)$-conjecture. These invariants determine the cohomological finiteness conditions of subgroups that contain the derived subgroup. We extend a known result for even Artin groups of FC-type, giving a sufficient condition for a character $χ:A_Γ\to\mathbb{R}$ to belong to $Σ^n(A_Γ,\mathbb{Z})$. We also prove some partial converses. As applications, we prove that the $Σ^1$-conjecture holds true when there is a prime $p$ that divides $l(e)/2$ for any edge with even label $l(e)>2$, we generalize to Artin groups the homological version of Bestvina-Brady theorem and we compute the $Σ$-invariants of all irreducible spherical and affine Artin groups and triangle Artin groups, which provide a complete classification of the $F_n$ and $FP_n$ properties of their derived subgroup.

math.GR↗