Related papers: The Geometric Invariants of Group Extensions Part …

In this paper, we compute the {\Sigma}^n(G) and {\Omega}^n(G) invariants when 1 \rightarrow H \rightarrow G \rightarrow K \rightarrow 1 is a short exact sequence of finitely generated groups with K finite. We also give sufficient conditions…

We compute the {\Omega}^1(G) invariant when 1 {\to} H {\to} G {\to} K {\to} 1 is a split short exact sequence. We use this result to compute the invariant for pure and full braid groups on compact surfaces. Applications to twisted conjugacy…

Let S be a generating set of a group G. We say that G has FINITE WIDTH relative to S if G=(S\cup S^{-1})^k for a suitable natural number k. We say that a group G is a group of FINITE C-WIDTH if G has finite width with respect to all…

For a finitely generated group $G$ we calculate the Bieri-Neumann-Strebel-Renz invariant $\Sigma^1(\X(G))$ for the weak commutativity construction $\X(G)$. Identifying $S(\X(G))$ with $S(\X(G) / W(G))$ we show $\Sigma^2(\X(G),\Z) \subseteq…

Let $G$ be a commutative affine algebraic group over a field $F$, and let $H \colon \mathrm{Fields}_{F} \to \mathrm{AbGrps}$ be a functor. A (homomorphic) $H$-invariant of $G$ is a natural transformation $\mathrm{Tors}(-, G) \to H$, where…

We generalize the theory of the second invariant cohomology group $H^2_{\rm inv}(G)$ for finite groups $G$, developed in [Da2,Da3,GK], to the case of affine algebraic groups $G$, using the methods of [EG1,EG2,G]. In particular, we show that…

Given a finitely-generated group G, and a finite group \Gamma, Philip Hall defined \delta_\Gamma to be the number of factor groups of G that are isomorphic to \Gamma. We show how to compute the Hall invariants by cohomological and…

In previous work, joint with Bux, Fluch, Marschler and Witzel, we proved that the braided Thompson groups are of type $\textrm{F}_\infty$. The proof utilized certain contractible cube complexes, which in this paper we prove are CAT(0). We…

Given a number field $F$, a finite group $G$ and an indeterminate $T$, {\it{a $G$-parametric extension over $F$}} is a finite Galois extension $E/F(T)$ with Galois group $G$ and $E/F$ regular that has all the Galois extensions of $F$ with…

If G is a semidirect product N by H with N normal and finitely generated then G has the property that every finite group is a quotient of some finite index subgroup of G if and only if one of N and H has this property. This has applications…

Let $G$ be an affine algebraic group over an algebraically closed field $k$ of characteristic zero. In this paper, we consider finite $G$-equivariant morphisms $F:X\to Y$ of irreducible affine $G$-varieties. First we determine under which…

We explore the topological full group [[G]] of an essentially principal etale groupoid G on a Cantor set. When G is minimal, we show that [[G]] (and its certain normal subgroup) is a complete invariant for the isomorphism class of the etale…

The authors classify the finite index subgroups of R. Thompson's group $F$. All such groups that are not isomorphic to $F$ are non-split extensions of finite cyclic groups by $F$. The classification describes precisely which finite index…

Let G be a reductive linear algebraic group over a field k. Let A be a finitely generated commutative k-algebra on which G acts rationally by k-algebra automorphisms. Invariant theory tells that the ring of invariants A^G=H^0(G,A) is…

J. Pevtsova and the author constructed a ``universal $p$-nilpotent operator" for an infinitesimal group scheme $G$ over a field $k$ of characteristic $p > 0$ which led to coherent sheaves on the scheme of 1-parameter subgroups of $G$…

If G is a finitely generated group, and A an algebraic group, then Hom(G,A) is a possibly reducible algebraic variety denoted by R_A(G). Here we define the profile function, P_d(R_A(G)), of the representation variety of G over A to be…

For every finite field F and every positive integer r, there exists a finite extension F' of F such that either SO(2r+1,F') or its simple derived group can be realized as a Galois group over Q. If the characteristic of F is 3 or 5 (mod 8),…

We fix a field $\kk$ of characteristic $p$. For a finite group $G$ denote by $\delta(G)$ and $\sigma(G)$ respectively the minimal number $d$, such that for any finite dimensional representation $V$ of $G$ over $\kk$ and any $v\in…

Let G be a simple, simply connected algebraic group defined over an algebraically closed field k of positive characteristic p. Let \sigma:G->G be a strict endomorphism (i. e., the subgroup G(\sigma) of \sigma-fixed points is finite). Also,…

The aim of the article is to show that there are many finite extensions of arithmetic groups which are not residually finite. Suppose $G$ is a simple algebraic group over the rational numbers satisfying both strong approximation, and the…