MathDB

Defect conjecture for Cayley graphs

For every constant Zp\mathbb{Z}_p-tower of Cayley graphs (Xn)n≥0(X_n)_{n\geq 0}, the defect δ(Xn)\delta(X_n) is eventually constant; that is, there exist N≥0N\geq 0 and dd such that δ(Xn)=d\delta(X_n)=d for all n≥Nn\geq N.

References

Primary source

arXiv

Additional references

algebraic-graph-theorygraph-theoryiwasawa-theory
Posted · edited

Progress summary

Refreshed
Claimed progress

A new paper reports proving the conjectured stability for several important families of Cayley-graph towers, but not for all such towers.

The defect conjecture predicts stability of the defect in suitable Cayley-graph towers. The general conjecture for arbitrary constant Zp\mathbb{Z}_p-towers remains outside the claimed result.

September 15, 2026 development

Debanjana Kundu, Katharina Müller, Alexis Newton, Chloe Stewart, and Yidi Wang report that the defect remains constant for several nontrivial families of Cayley-graph towers. They also compute Iwasawa invariants for associated Bowen–Franks groups. This is a substantial family-by-family advance, but it does not settle the arbitrary-tower conjecture, and the reported proofs are unverified here.

Current status (as of September 2026): Stability is claimed for several specified families, while the conjecture for arbitrary constant Zp\mathbb{Z}_p-towers remains open and the reported advance is unverified.

Sources

Solutions 0

No solutions have been posted yet.