Groups related to scissors-congruence groups. (English) Zbl 0674.55012

Lichtenbaum, Stephen

Algebraic \(K\)-theory and algebraic number theory, Proc. Semin., Honolulu/Hawaii 1987, Contemp. Math. 83, 151-157 (1989).

[For the entire collection see Zbl 0655.00010.]
This is an elementary treatment of various groups associated to the scissors congruence groups. The scissors congruence groups arose from an ancient geometric problem - the study of the volume function for polytopes. In dimension 3 or higher, an “exhaustion procedure” was used in place of a purely algebraic procedure. Recently, the problem has found delicate connections with many areas of mathematical investigations - number theory, invariants of flat bundles, algebraic K-theory, etc. The following works (to appear or have appeared) contain more information and more extensive references:
P. Cartier, Décomposition des polyèdres: Le point sur le Troisième Problème de Hilbert, Sémin. Bourbaki, 37e année, Vol. 1984/85, Exp. 646, Astérisque 133/134, 261-288 (1984; Zbl 0589.51032). C. H. Sah, Homology of classical Lie groups made discrete. III, J. Pure Appl. Algebra 56, 269-312 (1989). A. A. Beilinson, A. B. Goncharov, V. V. Schechtman, A. N. Varchenko, Aomoto dilogarithms, mixed Hodge structures and motivic cohomology of pairs of triangles on the plane (Preprint, 1988). D. Zagier, The remarkable dilogarithm, J. Math. Phys. Sci. 22, 131-145 (1988). D. Zagier, The Bloch-Wigner-Ramakrishnan polylogarithm function (Preprint, 1989).
The open problem mentioned at the end for finite fields has been studied by K.Hutchinson in his Ph. D. dissertation (1987).
The following paper provides an alternate proof of the Dehn-Sydler Theorem in the classical case of Euclidean space of dimension 3: J. L. Dupont and C. H. Sah, Homology of Euclidean motions groups made discrete (to appear in Acta Math.).

Reviewer: C.H.Sah

MSC:

55R35	Classifying spaces of groups and \(H\)-spaces in algebraic topology
20J99	Connections of group theory with homological algebra and category theory
51F20	Congruence and orthogonality in metric geometry
18F30	Grothendieck groups (category-theoretic aspects)
18F25	Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects)
57R20	Characteristic classes and numbers in differential topology
52Bxx	Polytopes and polyhedra
57R32	Classifying spaces for foliations; Gelfand-Fuks cohomology

Keywords:

Hilbert’s third problem; scissors congruence groups; Lie groups made discrete; dilogarithms; mixed Hodge structures; Dehn-Sydler Theorem

Citations:

Zbl 0655.00010; Zbl 0589.51032