Uniform conformal-volume conjecture for elliptic orbifolds

Let $n\leq 16$ , and let $\mathcal{O}$ be an elliptic $n$ -orbifold. Write $V_{c}(\mathcal{O})$ for its conformal volume. Uniform conformal-volume conjecture. There is a function $K(n)$ such that

V_{c}(\mathcal{O})\leq K(n).

A bound of this kind would provide the conformal-volume estimate needed to extend the paper's finiteness argument to the higher-dimensional range under discussion; the source does not establish the conjecture.

Equivalent formulations 1

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

Uniform conformal volume conjecture for elliptic orbifolds
Let $n$ be a dimension and let $\mathcal{O}$ be an elliptic $n$ -orbifold. Denote by $V_{PC}(\mathcal{O})$ its conformal volume in the sense used in the paper. Uniform conformal volume conjecture. There is a function $K(n)$ such that
$V_{PC}(\mathcal{O})\leq K(n).$
The conjecture would provide an alternative argument for the paper's finiteness theorem by giving a dimension-dependent bound on the relevant conformal volume.
source: Ian Agol, Mikhail Belolipetsky, Peter Storm and Kevin Whyte, “Finiteness of arithmetic hyperbolic reflection groups”, arXiv:math/0612132 (2006).

Sources & referencesView supporting material

Primary source

Ian Agol, “Finiteness of arithmetic Kleinian reflection groups”, arXiv:math/0512560 (2005).

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Equivalent formulations 1

Uniform conformal volume conjecture for elliptic orbifolds

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