Forester's lattices and small non-Leighton complexes

We construct two CW-complexes $K$ and $L$ admitting a common, but not finite common, covering, where $K$ is homeomorphic to a complex with a single 2-cell.

💡 Research Summary

The paper addresses the long‑standing question of whether a non‑Leighton pair of two‑dimensional CW‑complexes can be realized with only a single 2‑cell in one of the complexes. Leighton’s theorem guarantees that for finite graphs a common covering always yields a finite common covering, and many extensions of this result have been studied. In contrast, for two‑dimensional complexes the analogous statement fails: the first known examples required six 2‑cells in each complex, later reduced to four, then to two. Whether a pair with a single 2‑cell exists remained open.

The authors construct two CW‑complexes K and L that answer this question “almost”. Complex K is the standard one‑vertex, two‑cell presentation of the Baumslag–Solitar group BS(2,4)=⟨c,d | c²d = c⁴⟩. Topologically K is homeomorphic to a complex with a single 2‑cell, and therefore it is itself a Leighton‑type complex (it cannot belong to any non‑Leighton pair). By adding a single edge that splits one of the 2‑cells, they obtain a new complex K′ that contains two 2‑cells; this K′ will be paired with a more complicated complex L.

Complex L consists of two vertices (colored black and white), six edges (two loops c•, c∘ at each vertex, two “cross” edges y, z from black to white, and two edges t, t₁ from white to black), and four 2‑cells A, B, C, D with explicit attaching maps. The authors compute a presentation for π₁(L) and, using Schreier’s method, exhibit an index‑2 subgroup H that is isomorphic to the Baumslag–Solitar group BS(4,16)=⟨a,b | a⁴b = b¹⁶⟩. Since BS(2,4) and BS(4,16) are known to be incommensurable (no finite‑index subgroup of one is isomorphic to a finite‑index subgroup of the other), the fundamental groups of K and L are incommensurable. Consequently K and L cannot have a finite common covering, because any finite covering would induce a finite‑index subgroup common to both fundamental groups.

Despite the lack of a finite common cover, the universal coverings of K and L are isomorphic. Both are realized as the Baumslag–Solitar complex X₂,₄ introduced by Forester. X₂,₄ is built from a regular directed tree T in which each vertex has out‑degree 2 and in‑degree 4. Edges of T are labelled by a pair of bits (γ,δ)∈{0,1}², and the complex X₂,₄ is the Cartesian product T×ℝ equipped with a cell structure: vertices (v,i), horizontal edges ε_{v,i}, diagonal edges e_i, and 2‑cells D_{e,i} that fill the squares determined by the tree edges and the ℝ‑direction.

The covering map X₂,₄→K collapses all vertices of T to the single vertex of K, sends each horizontal edge to the generator c, and maps diagonal edges alternately to d or z depending on parity, thereby folding the two 2‑cells of K according to the parity of the ℝ‑coordinate. The covering X₂,₄→L respects the black/white coloring of T’s vertices, sends horizontal edges to the appropriate loop (c• or c∘), and maps diagonal edges to y, z, t, or t₁ according to the colour of the tail vertex and the δ‑label. The 2‑cells D_{e,i} are sent to A, B, C, or D depending on the colour and the parity of i+δ(e). The authors verify that these assignments respect attaching maps, thus establishing a genuine covering.

The main theorem therefore states: (i) K and L have isomorphic universal coverings (X₂,₄); (ii) they have no finite common covering because their fundamental groups are incommensurable; (iii) K is homeomorphic to a one‑2‑cell complex, while L is homeomorphic to a three‑2‑cell complex (its four 2‑cells can be merged). Moreover, π₁(K) is a one‑relator group, and π₁(L) is commensurable with a one‑relator group (BS(4,16)). This yields a new extremal example: a non‑Leighton pair where one member is essentially a single‑2‑cell complex, and the other is considerably larger.

The paper concludes by relating these findings to the Bridson–Shepherd theorem, which asserts that in a non‑Leighton pair neither fundamental group can be free or virtually free. The authors pose the open question of whether a non‑Leighton pair exists in which both fundamental groups are one‑relator groups. Their construction provides a partial answer, showing that one side can be a genuine one‑relator group while the other is only virtually so. The work thus advances the understanding of minimal non‑Leighton complexes and highlights the subtle role of Baumslag–Solitar groups and their commensurability properties in covering theory.