Séminaire Combinatoire et Théorie des Nombres ICJ

# Tessellations of hyperbolic 3-space and Bloch groups.

## by Rob de Jeu (Vrije Universiteit)

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

### Bât. Braconnier, salle Fokko du Cloux

#### ICJ, Université Lyon 1

Description

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.

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