Groups and polytopes
Abstract
In a series of papers the authors associated to an $L^2$-acyclic group $\Gamma$ an invariant $\mathcal{P}(\Gamma)$ that is a formal difference of polytopes in the vector space $H_1(\Gamma;\Bbb{R})$. This invariant is in particular defined for most 3-manifold groups, for most 2-generator 1-relator groups and for all free-by-cyclic groups. In most of the above cases the invariant can be viewed as an actual polytope. In this survey paper we will recall the definition of the polytope invariant $\mathcal{P}(\Gamma)$ and we state some of the main properties. We conclude with a list of open problems.
- Publication:
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arXiv e-prints
- Pub Date:
- November 2016
- DOI:
- 10.48550/arXiv.1611.01857
- arXiv:
- arXiv:1611.01857
- Bibcode:
- 2016arXiv161101857F
- Keywords:
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- Mathematics - Geometric Topology;
- Mathematics - Group Theory
- E-Print:
- 19 pages, 4 figures