Groups acting on trees with almost prescribed local action

We investigate a family of groups acting on a regular tree, defined by prescribing the local action almost everywhere. We study lattices in these groups and give examples of compactly generated simple groups of finite asymptotic dimension (actually one) not containing lattices. We also obtain examples of simple groups with simple lattices, and we prove the existence of (infinitely many) finitely generated simple groups of asymptotic dimension one. We also prove various properties of these groups, including the existence of a proper action on a CAT(0) cube complex.Comment: v2: 35 pages; argument slightly modified in 4.2.2; final versio

Compact presentability of tree almost automorphism groups

We establish compact presentability, i.e. the locally compact version of finite presentability, for an infinite family of tree almost automorphism groups. Examples covered by our results include Neretin's group of spheromorphisms, as well as the topologically simple group containing the profinite completion of the Grigorchuk group constructed by Barnea, Ershov and Weigel. We additionally obtain an upper bound on the Dehn function of these groups in terms of the Dehn function of an embedded Higman-Thompson group. This, combined with a result of Guba, implies that the Dehn function of the Neretin group of the regular trivalent tree is polynomially bounded.Comment: The results are extended to some almost automorphism groups of trees associated with closed regular branch groups. In particular we prove that the simple group (containing the profinite completion of the Grigorchuk group) constructed by Barnea, Ershov and Weigel, is compactly presente

Power-free values of the polynomial t_1...t_r - 1

Let k, r > 1 be two integers. We prove an asymptotic formula for the number of k-free values of the r variables polynomial t_1...t_r - 1 over the integral points of [1, x]^r.Comment: Final versio