Tessellations of hyperbolic 3-space and Bloch groups.

par Rob de Jeu (Vrije Universiteit)

mardi 14 mai 2019, 10:45 → 11:45 Europe/Paris

Bât. Braconnier, salle Fokko du Cloux (ICJ, Université Lyon 1)

Let $k$ be an imaginary quadratic number field with ring of integers $R$. We discuss how an ideal tessellation of hyperbolic 3-space on which $G{L}_2(R)$ acts gives rise to an explicit element $b$ of infinite order in the second Bloch group for $k$, and hence to an element $c$ in $K_3(k)$ modulo torsion, which is cyclic of infinite order.  The regulator of $c$ equals $-24{\zeta }_k^{\prime }(-1)$, and the Lichtenbaum conjecture for $k$ at $-1$ implies that a generator of $K_3(k)$ modulo torsion can be obtained by dividing $c$ by twice the order of $K_2(R)$. (The Lichtenbaum conjecture at $0$, because of the functional equation, amounts to the classical formula for the residue at $s=1$ of the zeta-function, involving the regulator of $R^{\ast }=K_1(R)$, the size of the torsion subgroup of $R^{\ast }$, and the class number of $R$.)

This division could be carried out explicitly in several cases by dividing $b$ in the second Bloch group.  The most notable case is that of $\mathbb{Q}(\sqrt{-303})$, where $K_2(R)$ has order 22.
                                                                                                                                                                                                                   
This is joint work with David Burns, Herbert Gangl, Alexander Rahm, and Dan Yasaki.