Related papers: Groups with cofinite Zariski topology and potentia…

Let $S$ be either a free group or the fundamental group of a closed hyperbolic surface. We show that if $G$ is a finitely generated residually-$p$ group with the same pro-$p$ completion as $S$, then two-generated subgroups of $G$ are free.…

The following question was asked by V. V. Bludov in The Kourovka Notebook in 1995: If a torsion-free group $G$ has a finite system of generators $a_1$, ..., $a_n$ such that every element of $G$ has a unique presentation in the form…

We prove a conjecture of Medvedev and Scanlon in the case of regular morphisms of semiabelian varieties. That is, if $G$ is a semiabelian variety defined over an algebraically closed field $K$ of characteristic $0$, and $\varphi\colon G\to…

It is a simple fact that a subgroup generated by a subset $A$ of an abelian group is the direct sum of the cyclic groups $\langle a\rangle$, $a\in A$ if and only if the set $A$ is independent. In [5] the concept of an $independent$ set in…

We discuss the decomposability of torsion-free abelian groups. We show that among computable groups of finite rank this property is $\Sigma^0_3$-complete. However, when we consider groups of infinite rank, it becomes $\Sigma^1_1$-complete,…

Given an aleph_1-free abelian group G we characterize the class C_G of all torsion abelian groups T satisfying Ext(G,T)=0 assuming the continuum hypothesis CH. Moreover, in Godel's constructable universe we prove that this characterizes C_G…

We find conditions which ensure that the topological complexity of a closed manifold $M$ with abelian fundamental group is nonmaximal, and see through examples that our conditions are sharp. This generalizes results of Costa and Farber on…

This paper describes some generalizations of the results presented in the book "Geometry of defining Relations in Groups" , of A.Yu.Ol'shanskii to the case of non-cyclic torsion-free hyperbolic groups. In particular, it is proved that for…

We prove a conjecture of Medvedev and Scanlon for endomorphisms of connected commutative linear algebraic groups $G$ defined over an algebraically closed field $\mathbb{k}$ of characteristic $0$. That is, if $\Phi\colon G\longrightarrow G$…

We generalize Hrushovski's Group Configuration Theorem to quasiminimal classes. As an application, we present Zariski-like structures, a generalization of Zariski geometries, and show that a group can be found there if the pregeometry…

We show that complex local systems with quasi-unipotent monodromy at infinity over a normal complex variety are Zariski dense in their moduli. v2: we waited for feedback and added a consequence of Alexandr Petrov's theorem. 3: we tightened…

Given a Tychonoff space $X$, let $F(X)$ and $A(X)$ be respectively the free topological group and the free Abelian topological group over $X$ in the sense of Markov. In this paper, we provide some topological properties of $X$ whenever one…

The Atiyah conjecture for a discrete group G states that the $L^2$-Betti numbers of a finite CW-complex with fundamental group G are integers if G is torsion-free and are rational with denominators determined by the finite subgroups of G in…

We prove that any countable discrete and torsion free subgroup of a general linear group over an arbitrary field or a similar subgroup of an almost connected Lie group satisfies the integral algebraic K-theoretic (split) Novikov conjecture…

An almost Abelian Lie group is a non-Abelian Lie group with a codimension 1 Abelian subgroup. We show that all discrete subgroups of complex simply connected almost Abelian groups are finitely generated. The topology of connected almost…

Assuming the abundance conjecture and the existence of a Zariski dense set of rational curves on terminal Calabi--Yau varieties, we show that a complex projective weakly special manifold $X$ with no rational curves is an \'etale quotient of…

We show that a surface group contained in a reductive real algebraic group can be deformed to become Zariski dense, unless its Zariski closure acts transitively on a Hermitian symmetric space of tube type. This is a kind of converse to a…

We study covers of the multiplicative group of an algebraically closed field as quasiminimal pregeometry structures and prove that they satisfy the axioms for Zariski-like structures presented in \cite{lisuriart}, section 4. These axioms…

Suppose $F$ is either a global field or a finitely generated extension of ${\mathbf Q}$, $A$ is an abelian variety over $F$, and $\ell$ is a prime not equal to the characteristic of $F$. Let $Z$ denote the center of the endomorphism algebra…

We show that every definable group G in an o-minimal structure is definably finitely generated. That is, G contains a finite subset that is not included in any proper definable subgroup. This provides another proof, and a generalization to…