Séminaire Combinatoire et Théorie des Nombres ICJ

Tessellations of hyperbolic 3-space and Bloch groups.

by Rob de Jeu (Vrije Universiteit)

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

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 $GL_2(R)$ 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 $-24 \zeta_k'(-1)$, 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 $R^*=K_1(R)$, the size of the torsion subgroup of $R^*$, 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.

Your browser is out of date!

Update your browser to view this website correctly. Update my browser now