Type-I Conjecture Section Heading

Definitions and mathematical statement

The only recoverable mathematical setup is: \[ G\text{ is a second countable locally compact group.} \] Second countability means that the topology of $G$ has a countable base. Local compactness means that the identity has a compact neighbourhood, and hence every point does by translation.

Unresolved mathematical statement. The supplied extraction contains the heading "A type I conjecture", the setup above, and the talk title "Hyperbolic groups of type I", attributed to Pierre-Emmanuel Caprace, with joint work by Mehrdad Kalantar and Nicolas Monod. It supplies no conjectural conclusion or quantified assertion beyond the setup. In particular, neither the type-I property nor hyperbolicity can be imposed on $G$ merely from these headings.

The supplied statement-verification note describes this extraction as a section heading and incomplete setup duplicated by a following complete conjecture. That following conjecture is absent from the supplied record. Consequently, no precise conjecture can be recovered here without additional source text. The record's open label and its assessment of a general locally compact-group conjecture are supplied curation, not a complete mathematical formulation or an independently verified status.

Short English statement

The record introduces a second countable locally compact group but omits the conjecture itself. Its type-I and hyperbolic-group headings do not determine the missing hypotheses or conclusion. This draft preserves the surviving setup and requires the missing source passage before a precise problem can be stated.

Sources

• Local UnsolvedMath registry, permanent ID $30004991$, source problem code catalogue entry, title "Type-I Conjecture Section Heading", in the Oberwolfach Reports Open Problems collection. The supplied record marks the statement unrecoverable and the problem open; these are registry labels.
• Supplied report citation: Groups and Dynamics: Topology, Measure, and Borel Structure (2023), Oberwolfach Reports, DOI 10.4171/owr/2022/3, pp. 107--169. https://doi.org/10.4171/owr/2022/3. The record separately gives proposed year 2022. The extracted talk title is "Hyperbolic groups of type I", attributed to Pierre-Emmanuel Caprace, joint work with Mehrdad Kalantar and Nicolas Monod. The supplied verification note says that the official report identifies the extraction as a heading and incomplete setup duplicated by a following complete conjecture; this verification claim has not been independently checked here.
• Caprace--Kalantar--Monod, A type I conjecture and boundary representations. https://arxiv.org/abs/2110.00190.
• Glasner, Boundary representations of locally compact hyperbolic groups. https://doi.org/10.1112/plms.70100.
• Literature assessment dated exactly 2026-08-22: the supplied curation describes the extraction as incomplete, reports substantial known hyperbolic and tree cases, and labels the general locally compact-group statement unresolved. This is supplied dated literature triage derived from the OWR curation, not an independently verified assessment. Its time-sensitive status and cited claims require independent verification.