Let F be a non-archimedean local field with residue class field k. Put G=GL2(F), Γ=PGL2(k) and let X denote the Bruhat–Tits tree of G. We construct a one-dimensional simplicial complex $\tilde X$, equipped with an action of G × Γ and with a G × Γ-equivariant simplicial projection $\pi: \tilde X\to X$ (for the trivial action of Γ on X). We prove that the cohomology with compact support $H^1_c(\tilde X\open C)$ contains nontrivial representations of G (in particular positive level supercuspidal representations).