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This paper is an expansion of my lecture for David Epstein’s birthday, which traced a
logical progression from ideas of Euclid on subdividing polygons to some recent
research on invariants of hyperbolic 3–manifolds. This “logical progression” makes a
good story but distorts history a bit: the ultimate aims of the characters in the story
were often far from 3–manifold theory.
We start in section 1 with an exposition of the current state of Hilbert’s 3rd
problem on scissors congruence for dimension 3. In section 2 we explain
the relevance to 3–manifold theory and use this to motivate the Bloch
group via a refined “orientation sensitive” version of scissors congruence.
This is not the historical motivation for it, which was to study algebraic
–theory
of
.
Some analogies involved in this “orientation sensitive” scissors congruence are not
perfect and motivate a further refinement in Section 4. Section 5 ties together various
threads and discusses some questions and conjectures.