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We prove that Thompson’s group  $$\mathrm {V}$$ is acyclic, answering a 1992 question of Brown in the positive. More generally, we identify the homology of the Higman–Thompson groups  $$\mathrm {V}_{n,r}$$ with the homology of the zeroth component of the infinite loop space of the mod  $$n-1$$ Moore spectrum. As  $$\mathrm {V}=\mathrm {V}_{2,1}$$ , we can deduce that this group is acyclic. Our proof involves establishing homological stability with respect to r, as well as a computation of the algebraic K-theory of the category of finitely generated free Cantor algebras of any type n.

Let $\mathcal{O}$ be an orbit in ℤ n of a finitely generated subgroup Λ of GL n (ℤ) whose Zariski closure Zcl(Λ) is suitably large (e.g. isomorphic to SL2). We develop a Brun combinatorial sieve for estimating the number of points on $\mathcal{O}$ at which a fixed integral polynomial is prime or has few prime factors, and discuss applications to classical problems, including Pythagorean triangles and integral Apollonian packings. A fundamental role is played by the expansion property of the “congruence graphs” that we associate with $\mathcal{O}$ . This expansion property is established when Zcl(Λ)=SL2, using crucially sum-product theorem in ℤ/qℤ for q square-free.

Let p be a prime number, let K be a field of characteristic not p, containing the p-th roots of unity, and let r≥1 be an integer. We compute the essential dimension of ℤ/p r ℤ over K (Theorem 4.1). In particular, i) We have edℚ(ℤ/8ℤ)=4, a result which was conjectured by Buhler and Reichstein in 1995 (unpublished). ii) We have edℚ(ℤ/p r ℤ)≥p r-1.