Boundaries of Groups and Kleinian Groups — Problem 89

Definitions and mathematical statement

Let $\Delta$ be a developable triangle of groups. Developability means that its local groups and inclusions are realized by stabilizers of an action of its complex-of-groups fundamental group $\Gamma=\pi_1(\Delta)$ on a simply connected development, with quotient the triangle. Thus $\pi_1(\Delta)$ is not the ordinary fundamental group of the underlying contractible triangle.

A discrete group has Kazhdan's Property $(T)$ if every unitary representation with almost invariant unit vectors has a nonzero invariant vector. Almost invariant means that, for every finite subset $F$ of the group and $\varepsilon>0$, some unit vector $v$ satisfies $\|\rho(g)v-v\|<\varepsilon$ for all $g\in F$.

Suppose every cell group of $\Delta$ has Property $(T)$, and every link graph $L$ in the development has \[ \lambda_1(L)>\frac12. \] Does it follow that $\Gamma$ has Property $(T)$? This preserves both the cell-group hypothesis and the strict spectral threshold.

The spectral normalization is missing. In a usual finite connected unweighted link convention, $\lambda_1$ is the first positive eigenvalue of the normalized graph Laplacian \[ (\mathcal Lf)(v)=f(v)-\frac1{\deg(v)}\sum_{w\sim v}f(w), \] considered with degree-weighted inner product. Weighted or infinite links need corresponding operator and spectral-gap definitions. The source gives no local-finiteness, connectivity, weighting, or normalization hypotheses, so the displayed finite-link convention is context, not an added source assumption. Its phrase "universal cover of T" has no separately defined space $T$; the development of $\Delta$ is the apparent intended reference and needs checking.

Short English statement

Must a developable triangle of Property $(T)$ groups have Property $(T)$ fundamental group when every development link has spectral gap greater than $1/2$? The link operator and development conventions are unspecified.

Sources

• Local UnsolvedMath registry, permanent ID 6200089, exact source code catalogue entry, "Boundaries of Groups and Kleinian Groups---Problem 89"; proposed year 2005, no proposer supplied. Registry status open, classification OPEN-TRIAGE.
• Kapovich, Problems on Boundaries of Groups and Kleinian Groups (2005), Problem 89, PDF page 22, https://www.math.ucdavis.edu/~kapovich/EPR/problems.pdf. Supplied normalized pdf-text extraction accessed July 29, 2026. No PDF was accessed here.
• The supplied summary relates the question to complexes of groups and Żuk-type spectral criteria, but supplies no precise theorem or publication and reports the triangle-of-groups assertion unestablished. That is a supplied open assessment, not an independent current-status check.
• Provenance flags current status and statement-reuse terms NEEDS_REVIEW; the document is publicly author-hosted.