On a theorem of Peter Scott

Abstract

We quantify Peter Scott's theorem that surface groups are locally extended residually finite (LERF) in terms of geometric data. In the process, we will quantify another result by Scott that any closed geodesic in a surface lifts to an embedded loop in a finite cover.

On the residual finiteness growths of particular hyperbolic manifold groups

“…More precisely, a group G is residually nite if for every non-identity element g ∈ G, there exists a nite index subgroup G of G such that g / ∈ G . Quantifying residual niteness, a concept rst introduced by K. Bou-Rabee in [6], refers to bounding the indexes of the nite index subgroups G in terms of algebraic data about G. In studying separability properties of the fundamental groups of manifolds, the bounds can also be given in terms of geometric data about the manifolds as in [20]. A quanti cation of residual niteness informs us on the minimal possible index of a subgroup G of G that separates g from the identity, and it has been studied for various classes of groups including free groups, surface groups, and virtually special groups (see for instance [6], [7], [8], [10], [12], [16], [17], [20], [23]).…”

“…Additionally, a quanti cation of residual niteness can serve as a foundation on which to build an approach to the quanti cation of stronger separability properties. In [20], the author presented a quanti cation of the residual niteness of hyperbolic surface groups in terms of geodesic length. The result was then used to make e ective a theorem of P. Scott [24] that these groups are also subgroup separable.…”

Building hyperbolic metrics suited to closed curves and applications to lifting simply

“…Gupta-Kapovich then gave a lower bound on f ρ (L) [7] that was recently improved upon by the second author. Combining the results of the second author [6] and third author [14] gives the following description of f ρ (L) found in [6, pp. 3]:…”

“…Our goal is to produce a complementary upper bound on f S (k) that is linear in k. Unfortunately, the upper bound on f ρ (L) from [14] does not immediately yield an upper bound on f S (k) as there exist arbitrarily long closed curves on a hyperbolic surface with few (or no) self-intersections. This issue is addressed by Theorem 1.4 above, in which a metric ρ is produced that is well suited to a comparison between ρ (γ) and the self-intersection i(γ, γ).…”

“…For a closed curve γ on S, let deg(γ) be the smallest degree of a cover where γ lifts simply. Recently, there has been an effort to prove effective versions of Scott's result by obtaining bounds on deg(γ) [6,7,14]. One of the main goals of this paper is to study the dependence of deg(γ) on two other notions of complexity; the geometric self-intersection number i(γ, γ) of the curve, and, fixing a complete hyperbolic metric ρ on S, the length ρ (γ).…”