Related papers: Zariski Closures and Subgroup Separability

We show that finite quasisimple groups of Lie type in characteristic $p$ with an irreducible representation of prime degree $r$ over a finite field of characteristic $p$ have orders bounded above by a function of $r$, independent of $p$. We…

We give the first part of a proof of Thurston's Ending Lamination conjecture. In this part we show how to construct from the end invariants of a Kleinian surface group a ``Lipschitz model'' for the thick part of the corresponding hyperbolic…

Answering a question of Dan Haran and generalizing some results of Aschbacher-Guralnick and Suzuki, we prove that given a set of primes pi, any finite group can be generated by a pi-subgroup and a pi'-subgroup. This gives a free product…

Let $1 \rightarrow H \rightarrow G \rightarrow \mathbb{Z} \rightarrow 1$ be an exact sequence of hyperbolic groups induced by a fully irreducible automorphism $\phi$ of the free group $H$. Let $H_1 (\subset H)$ be a finitely generated…

In this paper, we establish the Zariski decompositions of arithmetic R-divisors of continuous type on arithmetic surfaces and investigate several properties. We also develop the general theory of arithmetic R-divisors on arithmetic…

We study quasiminimal classes, i.e. abstract elementary classes (AECs) that arise from a quasiminimal pregeometry structure. For these classes, we develop an independence notion, and in particular, a theory of independence in $\M^{eq}$. We…

It is shown that for any finitely generated subgroups H and K of a free group F, and for any element g in F the double coset HgK is closed in the profinite topology of F.

It was shown in Part I that there exist strongly dense free subgroups in any semisimple algebraic group over a large enough field. These are nonabelian free subgroups all of whose subgroups are either cyclic or Zariski-dense. Here we show…

The polynomial method has been used recently to obtain many striking results in combinatorial geometry. In this paper, we use affine Hilbert functions to obtain an estimation theorem in finite field geometry. The most natural way to state…

In this paper we study the embedded topology of reducible plane curves having a smooth irreducible component. In previous studies, the relation between the topology and certain torsion classes in the Picard group of degree zero of the…

A framework is developed to describe the Zariski topologies on the prime and primitive spectra of a quantum algebra $A$ in terms of the (known) topologies on strata of these spaces and maps between the collections of closed sets of…

In the paper new criteria of existence and conjugacy of Hall subgroups of finite groups are given.

A group G is called subgroup conjugacy separable (abbreviated as SCS), if any two finitely generated and non-conjugate subgroups of G remain non-conjugate in some finite quotient of G. We prove that the free groups and the fundamental…

Suppose that all hyperbolic groups are residually finite. The following statements follow: In relatively hyperbolic groups with peripheral structures consisting of finitely generated nilpotent subgroups, quasiconvex subgroups are separable;…

We initiate a new, computational approach to a classical problem: certifying non-freeness of ($2$-generator, parabolic) M\"{o}bius subgroups of $\mathrm{SL}(2,\mathbb{Q})$. The main tools used are algorithms for Zariski dense groups and…

In this article, we study the Zariski closure of modular points in the two-dimensional universal deformation space when the residual Galois representation is reducible. Unlike the previous approaches in the residually irreducible case from…

Let $X$ be a non-positively curved cube complex with hyperbolic fundamental group. We prove that $\pi_1(X)$ has a non-free subgroup of infinite index unless $\pi_1(X)$ is either free or a surface group, answering questions of Gromov and…

The theory of covering spaces is often used to prove the Nielsen-Schreier theorem, which states that every subgroup of a free group is free. We apply the more general theory of semicovering spaces to obtain analogous subgroup theorems for…

Consider a one-ended word-hyperbolic group. If it is the fundamental group of a graph of free groups with cyclic edge groups then either it is the fundamental group of a surface or it contains a finitely generated one-ended subgroup of…

We characterize the Zariski topologies over an algebraically closed field in terms of general dimension-theoretic properties. Some applications are given to complex manifold and to strongly minimal sets.