Poly-freeness of even Artin groups of FC type

Poly-freeness in large even Artin groups

“…Finally, we will need the following result which is equivalent to Proposition 4.5 (3) in [9]. Given a set A, we will use #A to denote the cardinal of the set.…”

“…For example, rightangled Artin groups are poly-free: this fact was independently proved by Duchamp and Krob [7], Howie [12] and Hermiller and Šunić [8]. Martínez-Pérez, Paris and the author proved that even Artin groups of type FC are also poly-free [3]. In this paper, we add to the list of Artin groups known to be poly-free the family of large even Artin groups.…”

“…The Artin group associated to a triangle of type (i) is Z 3 , so it is poly-free. Artin groups associated to type (ii) triangles are even of type FC and so we know that it is also poly-free by [3] (In fact, these groups are of the form A 2 (2k 1 ) × Z so they are obviously poly-free). And Artin groups associated to triangles of type (iv) are large even Artin groups and thus they are also poly-free by Theorem 5.19.…”

“…An Artin group is called even if m e is an even number for all e ∈ E. There are not many results in the literature about even Artin groups (but see [3], [2] and [1]). However, we think that this family deserves more attention since they have remarkable properties some of them shared with the familiy of right-angled Artin groups.…”

The Farrell-Jones Conjecture for normally poly-free groups

“…Thus one might expect that our theorem can be used to verify the Farrell-Jones Conjecture for more Artin groups. Indeed, in [7], Blasco-Garcia, Martinez-Peréz, and Paris showed that even Artin groups of FC-type (see subsection 1.2 for the definition) are poly-free. A careful reading of their paper, in particular [7, Proposition 3.2], shows that even Artin groups of FC-type are actually normally poly-free.…”