Bloch group

The Bloch-Wigner function ${\displaystyle D_{2}(z)}$ , which is defined on ${\displaystyle \mathbb {C} \setminus \{0,1\}=\mathbb {C} P^{1}\setminus \{0,1,\infty \}}$ , has the following meaning: Let ${\displaystyle \mathbb {H} ^{3}}$ be 3-dimensional hyperbolic space and ${\displaystyle \mathbb {H} ^{3}=\mathbb {C} \times \mathbb {R} _{>0}}$ its half space model. One can regard elements of ${\displaystyle \mathbb {C} \cup \{\infty \}=\mathbb {C} P^{1}}$ as points at infinity on ${\displaystyle \mathbb {H} ^{3}}$. A tetrahedron, all of whose vertices are at infinity, is called an ideal tetrahedron. We denote such a tetrahedron by ${\displaystyle (p_{0},p_{1},p_{2},p_{3})}$ and its (signed) volume by ${\displaystyle \left\langle p_{0},p_{1},p_{2},p_{3}\right\rangle }$ where ${\displaystyle p_{0},\ldots ,p_{3}\in \mathbb {C} P^{1}}$ are the vertices. Then under the appropriate metric up to constants we can obtain its cross-ratio:

${\displaystyle \left\langle p_{0},p_{1},p_{2},p_{3}\right\rangle =D_{2}\left({\frac {(p_{0}-p_{2})(p_{1}-p_{3})}{(p_{0}-p_{1})(p_{2}-p_{3})}}\right)\ .}$

In particular, ${\displaystyle D_{2}(z)=\left\langle 0,1,z,\infty \right\rangle }$ . Due to the five terms relation of ${\displaystyle D_{2}(z)}$ , the volume of the boundary of non-degenerate ideal tetrahedron ${\displaystyle (p_{0},p_{1},p_{2},p_{3},p_{4})}$ equals 0 if and only if

${\displaystyle \left\langle \partial (p_{0},p_{1},p_{2},p_{3},p_{4})\right\rangle =\sum _{i=0}^{4}(-1)^{i}\left\langle p_{0},..,{\hat {p}}_{i},..,p_{4}\right\rangle =0\ .}$

In addition, given a hyperbolic manifold ${\displaystyle X=\mathbb {H} ^{3}/\Gamma }$ , one can decompose

${\displaystyle X=\bigcup _{j=1}^{n}\Delta (z_{j})}$

where the ${\displaystyle \Delta (z_{j})}$ are ideal tetrahedra. whose all vertices are at infinity on ${\displaystyle \partial \mathbb {H} ^{3}}$ . Here the ${\displaystyle z_{j}}$ are certain complex numbers with ${\displaystyle {\text{Im}}\ z>0}$ . Each ideal tetrahedron is isometric to one with its vertices at ${\displaystyle 0,1,z,\infty }$ for some ${\displaystyle z}$ with ${\displaystyle {\text{Im}}\ z>0}$ . Here ${\displaystyle z}$ is the cross-ratio of the vertices of the tetrahedron. Thus the volume of the tetrahedron depends only one single parameter ${\displaystyle z}$ . (Neumann & Zagier 1985) showed that for ideal tetrahedron ${\displaystyle \Delta }$ , ${\displaystyle vol(\Delta (z))=D_{2}(z)}$ where ${\displaystyle D_{2}(z)}$ is the Bloch-Wigner dilogarithm. For general hyperbolic 3-manifold one obtains

${\displaystyle vol(X)=\sum _{j=1}^{n}D_{2}(z)}$

by gluing them. The Mostow rigidity theorem guarantees only single value of the volume with ${\displaystyle {\text{Im}}\ z_{j}>0}$ for all ${\displaystyle j}$ .

Via substituting dilogarithm by trilogarithm or even higher polylogarithms, the notion of Bloch group was extended by Goncharov (Goncharov 1991) and Zagier (Zagier 1990). It is widely conjectured that those generalized Bloch groups B_n should be related to algebraic K-theory or motivic cohomology. There are also generalizations of the Bloch group in other directions, for example, the extended Bloch group defined by Neumann (Neumann 2004).

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