The Dodziuk conjecture over arbitrary fields

Let GG be the fundamental group of a finite-volume hyperbolic manifold of odd dimension, assume that GG is RFRS, and let βi(2)(G;K)\beta_i^{(2)}(G;\mathbb K) denote its iith L2L^2-Betti number with coefficients in a field K\mathbb K. Dodziuk conjecture. One has

βi(2)(G;K)=0\beta_i^{(2)}(G;\mathbb K)=0

for every ii and every field K\mathbb K.

This extends the known odd-dimensional real or complex hyperbolic L2L^2-acyclicity phenomenon to coefficients in arbitrary fields. The source gives no resolution status.

Sources & referencesView supporting material

Primary source

Dawid Kielak, “Virtual fibring of manifolds and groups”, arXiv:2510.01805 (2025).

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