The Single Obstruction

A group is coherent if every finitely generated subgroup is also finitely presented — meaning you can describe any piece of it with a finite set of rules. Checking this seems to require inspecting infinitely many subgroups, each potentially requiring its own finite presentation. An infinite verification task.

Fisher, Linton, and Sánchez-Peralta (arXiv:2603.16763) show that for a broad class of groups — virtually RFRS groups of cohomological dimension at most 2 — the entire question collapses to a single number: the second L²-Betti number. If it vanishes, the group is coherent. If it doesn’t, incoherent. No exceptions.

The L²-Betti numbers are topological invariants: they measure the “size” of homology using a von Neumann algebra framework. The second one, roughly, captures how much two-dimensional “stuff” the group’s classifying space carries that cannot be killed by finite-sheeted covers. When this number is zero, the group is virtually free-by-cyclic — it has a finite-index subgroup that decomposes as a free group extended by the integers.

What makes this striking is the compression ratio. Coherence is a property about all finitely generated subgroups. The second L²-Betti number is a single real number computed from the group’s overall topology. The infinite verification task reduces to reading off an invariant.

The result also delivers a global obstruction — the first known for any broad class. Previous incoherence proofs required constructing explicit counterexamples: finding a specific finitely generated subgroup that was not finitely presentable. Now the nonvanishing of one number suffices. You don’t need the counterexample. The invariant tells you it exists.

This is a recurring pattern in mathematics: what looks like it requires checking case-by-case turns out to be controlled by a single quantity computed at the level of the whole object. The local question has a global answer.