### A better proof of the Goldman-Parker conjecture.

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We prove that an Artin-Tits group of type $\tilde{C}$ is the group of fractions of a Garside monoid, analogous to the known dual monoids associated with Artin-Tits groups of spherical type and obtained by the “generated group” method. This answers, in this particular case, a general question on Artin-Tits groups, gives a new presentation of an Artin-Tits group of type $\tilde{C}$, and has consequences for the word problem, the computation of some centralizers or the triviality of the center. A key point of the proof...

2000 Mathematics Subject Classification: Primary 20F55, 13F20; Secondary 14L30.We construct invariant polynomials for the reflection groups [3, 4, 3] and [3, 3, 5] by using some special sets of lines on the quadric P1 × P1 in P3. Then we give a simple proof of the well known fact that the ring of invariants are rationally generated in degree 2,6,8,12 and 2,12,20,30.

Let $\U0001d524$ be a simple Lie algebra and ${\mathrm{\U0001d504\U0001d51f}}^{o}$ the poset of non-trivial abelian ideals of a fixed Borel subalgebra of $\U0001d524$. In [8], we constructed a partition ${\mathrm{\U0001d504\U0001d51f}}^{o}={\bigsqcup}_{\mu}{\mathrm{\U0001d504\U0001d51f}}_{\mu}$ parameterised by the long positive roots of $\U0001d524$ and studied the subposets ${\mathrm{\U0001d504\U0001d51f}}_{\mu}$. In this note, we show that this partition is compatible with intersections, relate it to the Kostant-Peterson parameterisation and to the centralisers of abelian ideals. We also prove that the poset of positive roots of $\U0001d524$ is a join-semilattice.

We show that the group of type-preserving automorphisms of any irreducible semiregular thick right-angled building is abstractly simple. When the building is locally finite, this gives a large family of compactly generated abstractly simple locally compact groups. Specialising to appropriate cases, we obtain examples of such simple groups that are locally indecomposable, but have locally normal subgroups decomposing non-trivially as direct products, all of whose factors are locally normal.