Cohomological dimension of just-non-polycyclic groups

DefinitionBase relators of the Kropholler group#✓fully proved

Let \(F\) be the free group on \(x, y\), viewed inside the free group on \(x, y, t\). The base relators are the elements \([[a, b], [c, d]]\) and \([a, b]^2\) for all \(a, b, c, d \in F\).

These elements generate \(F''(F')^2\) as a normal subgroup of \(F\), so they are relators for \(K = F/F''(F')^2\).

def baseRelators : Set (FreeGroup (Fin 3)) :=
  FreeGroup.map Fin.castSucc ''
    ({w : FreeGroup (Fin 2) | ∃ a b c d : FreeGroup (Fin 2), w = ⁅⁅a, b⁆, ⁅c, d⁆⁆} ∪
      {w : FreeGroup (Fin 2) | ∃ a b : FreeGroup (Fin 2), w = ⁅a, b⁆ ^ 2})

Used by 2

Used byThe Kropholler group · Rational cohomological dimension 4 of the Kropholler group (Kourovka 12.41)

DefinitionTwist relators of the Kropholler group#✓fully proved

In the free group on \(x, y, t\), the twist relators are the two elements \(t x t^{-1} y^{-1}\) and \(t y t^{-1} (x y)^{-1}\); they encode the relations \(t x t^{-1} = y\) and \(t y t^{-1} = x y\), so conjugation by \(t\) acts as the automorphism \(\varphi \colon x \mapsto y,\ y \mapsto x y\).

def twistRelators : Set (FreeGroup (Fin 3)) :=
  {FreeGroup.of 2 * FreeGroup.of 0 * (FreeGroup.of 2)⁻¹ * (FreeGroup.of 1)⁻¹,
    FreeGroup.of 2 * FreeGroup.of 1 * (FreeGroup.of 2)⁻¹ * (FreeGroup.of 0 * FreeGroup.of 1)⁻¹}

Used by 2

Used byThe Kropholler group · Rational cohomological dimension 4 of the Kropholler group (Kourovka 12.41)

AbbreviationThe Kropholler group#✓fully proved

The Kropholler group \(G\) is the group with generators \(x, y, t\) whose relators are the base relators, namely \([[a, b], [c, d]]\) and \([a, b]^2\) for all \(a, b, c, d\) in the free group \(F\) on \(x, y\), together with the twist relators \(t x t^{-1} y^{-1}\) and \(t y t^{-1} (x y)^{-1}\).

With \(K = F/F''(F')^2\) and \(\varphi\) the automorphism \(x \mapsto y\), \(y \mapsto x y\) of \(K\), this is the mapping torus of \(\varphi\), that is, the semidirect product \(K \rtimes \langle \varphi \rangle\) (\(\varphi\) has infinite order on \(K\)).

abbrev KrophollerGroup : Type :=
  PresentedGroup (baseRelators ∪ twistRelators)

Uses 2 · Used by 1

Statement usesBase relators of the Kropholler group · Twist relators of the Kropholler group

Used byRational cohomological dimension 4 of the Kropholler group (Kourovka 12.41)

DefinitionRational cohomological dimension 4 of the Kropholler group (Kourovka 12.41)#✓fully proved

Let \(G\) be the Kropholler group. This is the proposition that \(H^4(G, M) \neq 0\) for some \(\mathbb{Q}G\)-module \(M\), and \(H^n(G, M) = 0\) for every \(\mathbb{Q}G\)-module \(M\) and every integer \(n > 4\); that is, the cohomological dimension of \(G\) over \(\mathbb{Q}\) is \(4\).

This is the answer to Kourovka Notebook Problem 12.41 (P. H. Kropholler).

def rationalCohomologicalDimensionEqFour : Prop :=
  (∃ M : Rep ℚ KrophollerGroup, Nontrivial (groupCohomology M 4)) ∧
    ∀ (M : Rep ℚ KrophollerGroup) (n : ℕ), 4 < n → Subsingleton (groupCohomology M n)

Uses 3 · Used by 1

Statement usesThe Kropholler group · Base relators of the Kropholler group · Twist relators of the Kropholler group

Used byThe Kropholler group has rational cohomological dimension 4

TheoremThe Kropholler group has rational cohomological dimension 4#●ready to prove

Let \(G\) be the Kropholler group. Then \(H^4(G, M) \neq 0\) for some \(\mathbb{Q}G\)-module \(M\), and \(H^n(G, M) = 0\) for every \(\mathbb{Q}G\)-module \(M\) and every integer \(n > 4\). This resolves Kourovka Notebook Problem 12.41 by P. H. Kropholler, which asks whether this cohomological dimension is 3 or 4.

The Kourovka Notebook (21st issue) marks 12.41 solved: \(G\) is not constructible, so by Theorem III.8 of Bridson and Kropholler its cohomological dimension over \(\mathbb{Q}\) is its homological dimension plus 1, and the homological dimension is the Hirsch length 3 (U. Stammbach).

public theorem_wanted rationalCohomologicalDimensionEqFour_holds :
    rationalCohomologicalDimensionEqFour

Sources

• E. I. Khukhro and V. D. Mazurov (eds.), Unsolved Problems in Group Theory: The Kourovka Notebook, 21st issue, arXiv:1401.0300v49 (2026), Problem 12.41 (P. H. Kropholler); resolution: M. R. Bridson and P. H. Kropholler, J. Reine Angew. Math. 699 (2015), 217-243; U. Stammbach, J. London Math. Soc. (2) 2 (1970), 567-570, Theorem III.8 of Bridson-Kropholler (non-constructible groups: cd = hd + 1) and Stammbach (hd = Hirsch length = 3); the Kourovka Notebook (21st issue) marks 12.41 solved: it is 4.

Uses 1

Statement usesRational cohomological dimension 4 of the Kropholler group (Kourovka 12.41)

4 fully proved1 ready to prove

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