8. Algebraic properties
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We say a group is good in the sense of Serre if it has the same cohomology as its profinite completion.
Problem 8.1.
Are all Artin groups good in the sense of Serre? -
Problem 8.25.
Does the group ring of an Artin group have zero divisors? -
Problem 8.4.
Given a,b \in A, is there an n = n(a,b) such that \langle a^n, b^n \rangle is free or free abelian? -
Problem 8.45.
Compute the centralizer of an arbitrary element.
A more specific question: in a Euclidean Artin group, if two elements are contained in centers of spherical parabolic subgroups, when do the two elements commute? -
Problem 8.5.
Find a “good”/“natural” (e.g., finite dimensional) linear representation for Artin groups. -
Problem 8.6.
Which Artin groups have property R_\infty?-
Remark. [Ignat Soroko] Known to have property R_\infty: Spherical A_n, B_n, for n\ge2; D_4, I_2(m) for m\ge3, and the pure subgroups of all these; Euclidean: \tilde A_n, \tilde C_n; Certain classes of RAAGs (conjecturally all nonabelian RAAGs [Dekimpe–Senden]).
Known not to have property R_\infty: abelian RAAGs
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Cite this as: AimPL: Geometry and topology of Artin groups, available at http://aimpl.org/geomartingp.