Bloch group

In mathematics, the Bloch group is a cohomology group of the Bloch–Suslin complex, named after Spencer Bloch and Andrei Suslin. It is closely related to polylogarithm, hyperbolic geometry and algebraic K-theory.
Bloch–Wigner function.
The dilogarithm function is the function defined by the power series
It can be extended by analytic continuation, where the path of integration avoids the cut from 1 to +∞
The Bloch–Wigner function is related to dilogarithm function by
This function enjoys several remarkable properties, e.g.
The last equation is a variance of Abel's functional equation for the dilogarithm .
Definition.
Let "K" be a field and define formula_9 as the free abelian group generated by symbols ["x"]. Abel's functional equation implies that "D""2" vanishes on the subgroup "D" ("K") of "Z" ("K") generated by elements
Denote by "A" ("K") the factor-group of "Z" ("K") by the subgroup "D"("K"). The Bloch-Suslin complex is defined as the following cochain complex, concentrated in degrees one and two
then the Bloch group was defined by Bloch 
The Bloch–Suslin complex can be extended to be an exact sequence
This assertion is due to the Matsumoto theorem on K2 for fields.
Relations between K3 and the Bloch group.
If "c" denotes the element formula_15 and the field is infinite, Suslin proved the element "c" does not depend on the choice of "x", and
where GM("K") is the subgroup of GL("K"), consisting of monomial matrices, and BGM("K")+ is the Quillen's plus-construction. Moreover, let K3M denote the Milnor's K-group, then there exists an exact sequence
where K3("K")ind = coker(K3M("K") → K3("K")) and Tor("K"*, "K"*)~ is the unique nontrivial extension of Tor("K"*, "K"*) by means of Z/2.
Relations to hyperbolic geometry in three-dimensions.
The Bloch-Wigner function formula_18 , which is defined on formula_19 , has the following meaning: Let formula_20 be 3-dimensional hyperbolic space and formula_21 its half space model. One can regard elements of formula_22 as points at infinity on formula_20. A tetrahedron, all of whose vertices are at infinity, is called an ideal tetrahedron. We denote such a tetrahedron by formula_24 and its (signed) volume by formula_25 where formula_26 are the vertices. Then under the appropriate metric up to constants we can obtain its cross-ratio:
In particular, formula_28 . Due to the five terms relation of formula_18 , the volume of the boundary of non-degenerate ideal tetrahedron formula_30 equals 0 if and only if
In addition, given a hyperbolic manifold formula_32 , one can decompose
where the formula_34 are "ideal tetrahedra". whose all vertices are at infinity on formula_35 . Here the formula_36 are certain complex numbers with formula_37 . Each ideal tetrahedron is isometric to one with its vertices at formula_38 for some formula_39 with formula_37 . Here formula_39 is the cross-ratio of the vertices of the tetrahedron. Thus the volume of the tetrahedron depends only one single parameter formula_39 . showed that for ideal tetrahedron formula_43 , formula_44 where formula_18 is the Bloch-Wigner dilogarithm. For general hyperbolic 3-manifold one obtains
by gluing them. The Mostow rigidity theorem guarantees only single value of the volume with formula_47 for all formula_48 .
Generalizations.
Via substituting dilogarithm by trilogarithm or even higher polylogarithms, the notion of Bloch group was extended by Goncharov and Zagier . It is widely conjectured that those generalized Bloch groups Bn 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 .