Finiteness of Hyperbolic-Volume-Generated Homology Seminorms

Definitions and mathematical statement

The supplied source asks: Is the functorial seminorm on singular homology generated by the volumes of oriented closed connected hyperbolic manifolds finite? The record does not specify the coefficient ring, degree range, or construction meant by "generated." The following gives a precise conditional formulation using real coefficients and generation by finite linear combinations of images of fundamental classes; these conventions require confirmation against the cited source.

Fix an integer $n\geq 2$. For a topological space $X$, let $H_n(X;{\mathbb{R}})$ denote singular homology with real coefficients. An extended functorial seminorm in degree $n$ assigns a function \[ p_X:H_n(X;{\mathbb{R}})\longrightarrow[0,\infty] \] to each $X$, with $p_X(0)=0$, the triangle inequality, absolute homogeneity for nonzero real scalars, and \[ p_Y\left(H_n(f;{\mathbb{R}})(\alpha)\right)\leq p_X(\alpha) \] for every continuous map $f:X\to Y$ and every $\alpha\in H_n(X;{\mathbb{R}})$. It is finite if $p_X(\alpha)<\infty$ for every $X$ and $\alpha$; this does not require a uniform bound over all classes.

Under the conditional generation convention above, let $M$ range over oriented, closed, connected hyperbolic $n$-manifolds, where closed means compact without boundary and the hyperbolic metric is normalized to sectional curvature $-1$. Write $[M]_{{\mathbb{R}}}\in H_n(M;{\mathbb{R}})$ for its fundamental class and $\operatorname{Vol}(M)$ for its Riemannian volume. Define \[ p^{\mathrm{hyp}}_{n,X}(\alpha)= \inf\left\{ \sum_{j=1}^{k}|a_j|\operatorname{Vol}(M_j) \;\middle|\; \begin{gathered} k\in{\mathbb{Z}}_{\geq0},\quad a_j\in{\mathbb{R}},\\ f_j:M_j\to X\text{ continuous},\\ \alpha=\sum_{j=1}^{k}a_jH_n(f_j;{\mathbb{R}})([M_j]_{{\mathbb{R}}}) \end{gathered} \right\}. \] The empty sum represents zero, and the infimum of an empty set of representations is $+\infty$.

With these conventions, the finiteness question in a fixed degree is \[ \forall X\text{ topological space}\quad \forall\alpha\in H_n(X;{\mathbb{R}}),\qquad p^{\mathrm{hyp}}_{n,X}(\alpha)<\infty\;? \] The supplied record does not determine whether this is intended for each $n\geq2$, for a particular degree, or with another coefficient or generation convention. Accordingly, this conditional formulation is not asserted to be a verified reconstruction of the source's intended construction.

Short English statement

The question asks whether the homology seminorm generated by hyperbolic manifold volumes takes a finite value on every homology class in its intended domain. A precise formulation using real coefficients and finite sums of mapped fundamental classes is given conditionally above. The source record leaves the coefficients, degree range, and generation convention unspecified. Its label open is supplied curation, not an independently verified status.

Sources

• Supplied local UnsolvedMath registry record: permanent ID $30001813$, source problem code catalogue entry, collection Oberwolfach Reports Open Problems. No individual proposer is identified in the record.
• Arithmetic Groups vs. Mapping Class Groups: Similarities, Analogies and Differences (2011), Oberwolfach Reports, pp. 1637--1708, DOI 10.4171/owr/2011/30. Supplied source and literature reference, labeled "original Oberwolfach report": https://doi.org/10.4171/owr/2011/30.
• Supplied literature reference labeled "Loh--functorial seminorms": https://loeh.app.uni-regensburg.de/preprints/funcseminorms.pdf. The record supplies no further bibliographic details for this reference.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open and states that it was explicitly posed in the report and remains described as open in later treatments. This is supplied dated curation, not an independently verified literature assessment. The background expressly requests independent verification of time-sensitive status and cited claims.
• The registry marks the statement as "exact" and reports that its mathematical content was checked against the official report, with notation, context, and question form normalized. These are supplied source-verification claims; no independent source consultation is claimed here.