Linear representations of random groups

We show that for a fixed [Formula: see text], Gromov random groups with any density [Formula: see text] have no nontrivial degree [Formula: see text] representations over any field, a.a.s. This is especially interesting in light of the results of Agol, Ollivier and Wise that when [Formula: see text] such groups have a faithful linear representation over [Formula: see text], a.a.s.

REDUCIBILITY OF THE COHEN–WALES REPRESENTATION OF THE ARTIN GROUP OF TYPE Dn

We construct a linear representation of the CGW algebra of type Dn. This representation has degree n2 - n, the number of positive roots of a root system of type Dn. We show that the representation is generically irreducible, but that when the parameters of the algebra are related in a certain way, it becomes reducible. As a representation of the Artin group of type Dn, this representation is equivalent to the faithful linear representation of Cohen–Wales. We give a reducibility criterion for this representation as well as a conjecture on the semisimplicity of the CGW algebra of type Dn. Our proof is computer-assisted using Mathematica.

Noncomplete affine structures on Lie algebras of maximal class

Every affine structure on Lie algebra𝔤defines a representation of𝔤inaff(ℝn). If𝔤is a nilpotent Lie algebra provided with a complete affine structure then the corresponding representation is nilpotent. We describe noncomplete affine structures on the filiform Lie algebraLn. As a consequence we give a nonnilpotent faithful linear representation of the 3-dimensional Heisenberg algebra.

Faithful linear representations of certain free nilpotent groups

Brian Hartley asked me whether a free (nilpotent of class 2 and exponent p2)-group of countable rank has a faithful linear representation of finite degree, p here being a prime of course. The answer is yes. The point is that this then yields via work of F. Leinen and M. J. Tomkinson, see [3,3.6] an image of a linear p-group, which is not even finitary linear. The question of which relatively free groups have faithful linear representations dates back at least to work of W. Magnus in the 1930's, see [4, pp. 33, 34 and the final comment on p. 40] for a discussion of this. Our construction, which works more generally, is a further contribution. We write ℜc for the variety of nilpotent groups of class at most c and (ℭ9 for the variety of groups of exponent dividing q.