The outer automorphism groups of three classes of groups

The theory of outer automorphism groups allows us to better understand groups through their symmetries, and in this thesis we approach outer automorphism groups from two directions. In the first direction we start with a class of groups and then classify their outer automorphism groups. In the other direction we start with a broad class of groups, for example finitely generated groups, and for each group Q in this class we construct a group G_Q such that Q is related, in a suitable sense, to the outer automorphism group of G_Q. We give a list of 14 groups which precisely classifies the outer automorphism groups of one-ended two-generator, one-relator groups with torsion. We also describe the outer automorphism groups of such groups which have more than one end. Combined with recent algorithmic results of Dahmani–Guirardel, this work yields an algorithm to compute the outer automorphism group of a two-generator, one-relator group with torsion. We prove a technical theorem which, in a certain sense, writes down a specific subgroup of the outer automorphism group of a particular kind of HNN-extension. We apply this to prove two main results. These results demonstrate a universal property of triangle groups and are as follows. Fix an arbitrary hyperbolic triangle group H. If Q is a finitely generated group then there exists an HNN-extension G_Q of H such that Q embeds with finite index into the outer automorphism group of G_Q. Moreover, if Q is residually finite then G_Q can be taken to be residually finite. Secondly, fix an equilateral triangle group H = ⟨a, b; a^i, bi, (ab)^i⟩ with i > 9 arbitrary. If Q is a countable group then there exists an HNN-extension G_Q of H such that Q is isomorphic to the outer automorphism group of G_Q. The proof of this second main result applies a theory of Wise underlying his recent work leading to the resolution of the virtually fibering and virtually Haken conjectures. We prove a technical theorem which, in a certain sense, writes down a specific subgroup of the outer automorphism group of a semi-direct product

Local-global invariants of finite and infinite groups: around Burnside from another side

This expository essay is focused on the Shafarevich-Tate set of a group $G$. Since its introduction for a finite group by Burnside, it has been rediscovered and redefined more than once. We discuss its various incarnations and properties as well as relationships (some of them conjectural) with other local-global invariants of groups.Comment: 18 page

Sha-rigidity of Chevalley groups over local rings

We prove that every locally inner endomorphism of a Chevalley group (or its elementary subgroup) over a local ring with an irreducible root system of rank >1 (with 1/2 for the systems A_2, F_4, B_l, C_l and with 1/3 for the system G_2) is inner, so that all these groups are Sha-rigid.Comment: 14 page

Regular Maps on Surfaces with Large Planar Width

AbstractA map is a cell decomposition of a closed surface; it is regular if its automorphism group acts transitively on the flags, mutually incident vertex-edge-face triples. The main purpose of this paper is to establish, by elementary methods, the following result: for each positive integer w and for each pair of integersp≥ 3 and q≥ 3 satisfying 1/p+ 1/q≤ 1/2, there is an orientable regular map with face-size p and valency q such that every non-contractible simple closed curve on the surface meets the 1-skeleton of the map in at least w points. This result has several interesting consequences concerning maps on surfaces, graphs and related concepts. For example, MacBeath’s theorem about the existence of infinitely many Hurwitz groups, or Vince’s theorem about regular maps of given type (p, q), or residual finiteness of triangle groups, all follow from our result

On a geometric description of Gal(Qˉp/Qp)Gal(\bar{\bf Q}_p/{\bf Q}_p) and a p-adic avatar of GT^\hat{GT}

We develop a $p$-adic version of the so-called Grothendieck-Teichm\"uller theory (which studies $Gal(\bar{\bf Q}/{\bf Q})$ by means of its action on profinite braid groups or mapping class groups). For every place $v$ of $\bar{\bf Q}$, we give some geometrico-combinatorial descriptions of the local Galois group $Gal(\bar{\bf Q}_v/{\bf Q}_v)$ inside $Gal(\bar{\bf Q}/{\bf Q})$. We also show that $Gal(\bar{\bf Q}_p/{\bf Q}_p)$ is the automorphism group of an appropriate $\pi_1$-functor in $p$-adic geometry.Comment: version to appear in Duke Math.

Absolute profinite rigidity and hyperbolic geometry

We construct arithmetic Kleinian groups that are profinitely rigid in the absolute sense: each is distinguished from all other finitely generated, residually finite groups by its set of finite quotients. The Bianchi group $\mathrm{PSL}(2,\mathbb{Z}[\omega])$ with $\omega^2+\omega+1=0$ is rigid in this sense. Other examples include the non-uniform lattice of minimal co-volume in $\mathrm{PSL}(2,\mathbb{C})$ and the fundamental group of the Weeks manifold (the closed hyperbolic $3$-manifold of minimal volume).Comment: v2: 35 pages. Final version. To appear in the Annals of Mathematics, Vol. 192, no. 3, November 202