I think the names of Johan Louis Dupont ( http://home.imf.au.dk/dupont/ ), Chih-Han Sah (1934--1997) (SUNY Stony Brook) are appropriate.
Dupont's homepage contains related articles e.g.
Homology of O(n) and O^1(1,n) made discrete; an application of edgewise subdivision
by J. Dupont, M. Bökstedt and Morten Brun, J. Pure Appl. Algebra, 123 (1998), 131-152.
and many of his papers are in arXiv, e.g.
Regulators and characteristic classes of flat bundles
Johan Dupont, Richard Hain, Steven Zucker
The Lie groups made discrete arise in algebraic K-theory,
for small n the groups K_n are related to "scissors congruence groups" (as far as I understand),
see let me quote http://reh.math.uni-duesseldorf.de/~topologie/scissors/
"The basic reference for the school is the monograph
[D] J.L. Dupont, Scissors congruences, group homology and characteristic classes, World Scientific.
Further references occuring below are
[M] Milnor, On the homology of Lie groups made discrete. Commentarii Mathematici Helvetici, Vol. 58, No. 1, 72--85, 1983
[S] Suslin, A.A., Algebraic K-theory of fields. Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986), 222--244, Amer. Math. Soc., Providence, RI, 1987"
And further quote:
Two polytopes in euclidean n-space are called scissors congruent if they can be subdivided into finitely many pieces such that each piece in the first polytope is congruent to exactly one piece in the second polytope.
Elementary geometric considerations show that polytopes in the plane are scissors congruent if and only if they have the same area. Hilbert's 3rd problem was the question whether volume determines the scissors congruence class also in 3-space. The answer was given by Max Dehn almost immediately: In 1900, he described an invariant with values in R ⊗Z R/Z which shows that the answer is no. Only 1965 J. P. Sydler proved that volume and Dehn invariant together determine the scissors congruence class in 3-space. Higher dimensional analogues are still unsolved. There are variants for spherical and hyperbolic geometry, which are open even in dimension 3.
From a modern point of view, these classical questions are closely related to the computation of the homology of Lie groups considered as discrete groups. Furthermore there are interesting connections to deep questions about the algebraic K-theory of the complex numbers."
P. Cartier, Décomposition des polyèdres : le point sur le troisième problème de Hilbert, Séminaire Bourbaki, 1984-85, n° 646, p. 261—288.
Hilbert’s 3rd problem and Invariants of 3-manifolds by
Walter Neumann, G&T Monographs, 1998.