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In this paper by Robert Young, the author defines

We define a riemannian simplicial complex to be a simplicial complex with a metric which gives each simplex the structure of a riemannian manifold with corners. We say that such a complex is quasi-conformal (or that the complex is a QC complex ) if there is a $c$ such that the riemannian metric on each simplex is $c$-bilipschitz equivalent to a scaling of the standard simplex.

I have some problem understanding the next paragraph:

QC complexes are a compromise between the rigidity of simplicial complexes and the freedom of riemannian simplicial complexes. A key feature of simplicial complexes is that curves and cycles can be approximated by simplicial curves and cycles. This is not true in riemannian simplicial complexes, but it holds in QC complexes.

What confuses me is that in the definition of Riemannian simplicial complex one starts with a simplicial complex. But the paragraph below says that a Riemannian simplicial complex is not a simplicial complex.

Can anyone recommend additional references to both riemmanian simplicial complexes and QC complexes?

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    $\begingroup$ Robert is talking about the metric properties of these complexes; the rigidity he refers to comes from endowing each simplex with the metric coming from the standard simplex in $\mathbb{R}^n$. $\endgroup$ Mar 18, 2015 at 23:11
  • $\begingroup$ @AndyPutman oh, that clears things up. thanks. $\endgroup$ Mar 18, 2015 at 23:14
  • $\begingroup$ @AndyPutman so in short: simplical complex -> simplices have metric from standard simplex; riemannian simplicial complex -> simplices have riemannian metric that makes each simplex into a manifold; QC comples -> simplices have metric which is almost like that of standard simplex (up to some scaling)? $\endgroup$ Mar 18, 2015 at 23:18
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    $\begingroup$ Yes, I believe that is correct. $\endgroup$ Mar 18, 2015 at 23:28

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