Coarse Baum–Connes Failure without Embedded Expanders

Definitions and mathematical statement

Write $(M,d,\mu)$ for the compact metric probability space in the supplied record. Let $G$ be a finitely generated group acting on $M$ by measure-preserving Lipschitz homeomorphisms. Ergodicity means that every measurable set invariant under the action modulo null sets has measure zero or one. Choose a finite symmetric generating set $S\subseteq G$. Put \[ L^2_0(M,\mu)=\left\{f\in L^2(M,\mu):\int_M f\,{\,\mathrm{d}}\mu=0\right\}, \qquad U_s f(x)=f(s^{-1}x). \] The action has a spectral gap if there is $\kappa>0$ such that \[ \sum_{s\in S}\|U_s f-f\|_2^2\geq \kappa\|f\|_2^2 \quad\text{for every }f\in L^2_0(M,\mu). \]

A standard continuous interpretation of the warped cone $\mathcal O_G(M)$ has underlying set $M\times[1,\infty)$. Its metric is the infimum of the total costs of finite chains of the following moves: a vertical move $(x,t)\to(x,u)$ costs $|t-u|$, a horizontal move $(x,t)\to(y,t)$ costs $t\,d(x,y)$, and a generator move $(x,t)\to(sx,t)$, with $s\in S$, costs one. The supplied record does not specify whether its notation denotes this continuous cone, a family of levels with their intrinsic warped metrics, or a coarse disjoint union of discretized levels. This convention requires confirmation from the cited report.

For clarity, a standard expander sequence consists of finite connected graphs $(\Gamma_n)$ with vertex sets $V_n$, uniformly bounded degrees, $|V_n|\to\infty$, and a constant $h>0$ such that \[ |\partial_E A|\geq h|A| \quad\text{whenever }A\subseteq V_n,\quad 0<|A|\leq |V_n|/2, \] where $\partial_E A$ consists of edges having exactly one endpoint in $A$. Give each graph its shortest-path metric. Under the coarse-disjoint-union interpretation, such a sequence is coarsely embedded in a metric space $X$ if a coarse disjoint union $Y=\bigsqcup_n V_n$ admits a map $F:Y\to X$ and nondecreasing functions $\rho_-,\rho_+:[0,\infty)\to[0,\infty)$, with $\rho_-(r)\to\infty$ as $r\to\infty$, satisfying \[ \rho_-(d_Y(a,b))\leq d_X(F(a),F(b))\leq\rho_+(d_Y(a,b)) \quad(a,b\in Y). \] Here the metric on $Y$ restricts to each graph metric and separates distinct components at distances tending to infinity. The record does not specify whether this interpretation or uniform coarse embeddings of the individual graphs is intended.

In the usual complex, coefficient-free formulation, the coarse Baum--Connes conjecture for a space $X$, when this formulation applies, asserts that the coarse assembly map \[ \mu_X:KX_*(X)\longrightarrow K_*(C^*(X)) \] is an isomorphism in both degrees, where $KX_*(X)$ denotes coarse $K$-homology and $C^*(X)$ the Roe algebra. Failure means that this map is not an isomorphism. The source record does not specify the assembly-map variant, the discretization, or the geometric hypotheses needed for its intended formulation; the displayed convention therefore does not resolve those source questions.

Supplied conjecture and existence question. Subject to these unresolved conventions, the extracted targets are:

1. For every finitely generated group $G$ and every ergodic, measure-preserving action by Lipschitz homeomorphisms on a compact metric probability space $M$, if the action has a spectral gap, must $\mathcal O_G(M)$ fail the coarse Baum--Connes conjecture?
2. Does there exist a group and action satisfying all of those hypotheses for which $\mathcal O_G(M)$ contains no coarsely embedded sequence of expanders?

The original wording restricts the first conjecture to warped cones "in the stated construction," but that construction is not reproduced in the supplied record. Its clean statement supplies the hypotheses above; the equivalence of these scopes remains unresolved. No freeness, non-atomicity, infinitude of $M$, full support of $\mu$, or metric-measure regularity assumption is supplied, and none is added here. In particular, the spectral-gap inequality alone does not exclude $L^2_0(M,\mu)=\{0\}$. These missing qualifications prevent treating this draft as a fully determined formulation of the original report's problem.

Short English statement

The supplied problem asks whether every spectral-gap warped cone in the intended construction violates the coarse Baum--Connes conjecture, and whether some such cone contains no coarsely embedded expander sequence. The construction, embedding convention, and assembly-map formulation require confirmation. The registry labels the problem partially solved; this is supplied dated curation, not an independently verified mathematical conclusion.

Sources

• Local UnsolvedMath registry, permanent ID $30002761$, title: Coarse Baum--Connes Failure without Embedded Expanders; source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The supplied record gives proposed year 2015 and names no individual proposer.
• Supplied report citation: Geometric Topology (2015), Oberwolfach Reports, pp. 187--233, DOI 10.4171/owr/2015/3. https://doi.org/10.4171/owr/2015/3. The supplied background describes Oberwolfach Reports as records of workshops and their open-problem discussions. No exact problem page or speaker attribution is supplied.
• The registry labels the statement corrected_verified and says it was checked against the official report, preserving its hypotheses, notation, and target. This is a supplied verification claim; no independent source consultation is asserted here.
• Supplied literature assessment, checked exactly 2026-08-21: status partially_solved. The assessment reports coarse Baum--Connes failure for important spectral-gap warped cones and characterizations of expander graph models under regularity assumptions, while describing the unrestricted extracted existence question as not fully settled. This is supplied dated curation, not independently verified; the background expressly requests independent verification of time-sensitive status and cited claims.
• Supplied literature source: Sawicki, warped cones violating coarse Baum--Connes. https://www.impan.pl/~dsawicki/files/counterexamples.pdf.
• Supplied literature source: Kitsios, Schick, and Vigolo, warped cones. https://arxiv.org/abs/2504.21811.
• Supplied literature source: Sawicki, super-expanders and warped cones. https://doi.org/10.5802/aif.3373.