Interval Structure of Cohomology-Vanishing Exponents

Definitions and mathematical statement

Let $\sigma$ be a fixed continuous homomorphism as in the supplied statement, \[ \sigma:G\longrightarrow \operatorname{Aut}(X,[\mu])\ltimes L^0(X,\mu;\{-1,1\}). \] Here $[\mu]$ denotes the measure class of $\mu$; under the usual interpretation, $\operatorname{Aut}(X,[\mu])$ consists of invertible measurable transformations preserving this measure class, identified modulo equality almost everywhere, and $L^0(X,\mu;\{-1,1\})$ consists of measurable sign functions identified modulo equality almost everywhere. For each finite real exponent $p>0$, let $\pi_{p,\sigma}$ denote the induced linear isometric representation on $L^p(X,\mu)$ specified by the source. The same homomorphism $\sigma$ is held fixed as $p$ varies.

To make the cohomological target explicit, under the continuous, unreduced interpretation of first cohomology, put $E_p=L^p(X,\mu)$ and define \[ \begin{aligned} Z^1(G,\pi_{p,\sigma}) &=\{b:G\to E_p:\ b\text{ is continuous and }\\[-2pt] &\hspace{35mm}b(gh)=b(g)+\pi_{p,\sigma}(g)b(h)\text{ for all }g,h\in G\},\\ B^1(G,\pi_{p,\sigma}) &=\{b_v:\ b_v(g)=\pi_{p,\sigma}(g)v-v,\ v\in E_p\},\\ H^1(G,\pi_{p,\sigma}) &=Z^1(G,\pi_{p,\sigma})/B^1(G,\pi_{p,\sigma}). \end{aligned} \] This definition presupposes that the induced representations have the continuity needed for these coboundaries to be continuous. Vanishing means that every continuous cocycle is a coboundary; no closure of $B^1$ is taken.

The question is whether, for every datum covered by the source, the set \[ S_\sigma=\{p\in(0,\infty):H^1(G,\pi_{p,\sigma})=0\} \] is an interval. Explicitly, is it always true that \[ \begin{gathered} 0<p_0<q<p_1<\infty,\\ H^1(G,\pi_{p_0,\sigma})=0, \qquad H^1(G,\pi_{p_1,\sigma})=0 \quad\Longrightarrow\quad H^1(G,\pi_{q,\sigma})=0? \end{gathered} \] This order-convex formulation permits the empty set and singletons and imposes no endpoint inclusion condition.

The supplied record does not specify the class of topological groups $G$, hypotheses on $(X,\mu)$, the scalar field of $L^p$, the topology on the target of $\sigma$, or the precise construction and continuity assumptions for the induced representations. It also does not explicitly specify the cohomology convention; the continuous, unreduced convention above therefore requires confirmation. The range $0<p<1$ is retained: these spaces generally have a quasi-norm rather than a norm, so a Banach-space-only interpretation would not cover the stated question. These unresolved qualifications must be checked before this becomes a fully specified statement.

Short English statement

For one fixed homomorphism $\sigma$, must vanishing of first cohomology at two positive exponents force vanishing at every exponent between them? The supplied record presents this as a refined form of Drutu's conjecture and labels it open. Its analytic hypotheses and cohomology convention require confirmation.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30005041$, title Interval Structure of Cohomology-Vanishing Exponents, exact source problem code catalogue entry, in Oberwolfach Reports Open Problems. No individual proposer is supplied; the proposed year is recorded as 2022.
• Supplied report citation: Geometric Structures in Group Theory (2023), Oberwolfach Reports, DOI 10.4171/owr/2022/11, pp. 517--576. https://doi.org/10.4171/owr/2022/11. The background describes these reports as documenting mathematical workshops and their open-problem discussions. The supplied contextual passage calls this question a refined form of Drutu's conjecture; it does not supply that conjecture's full formulation.
• Supplied literature reference: Isometric actions on Lp spaces. https://arxiv.org/abs/2001.02490.
• Supplied literature reference: Oberwolfach report. https://doi.org/10.4171/owr/2022/11.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem open and states that the cited work explicitly leaves the general fixed-representation interval question unsettled. 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 labels the statement corrected_verified and records that the official report was checked directly and the statement restored while removing extraction artifacts. These are supplied verification claims, not verification performed for this draft.