Related papers: Moufang sets of finite Morley rank of odd type

Suppose $\mathbb{F}$ is a field and let $\mathbf{a} := (a_1, a_2, \dotsc)$ be a sequence of non-zero elements in $\mathbb{F}$. For $\mathbf{a}_n := (a_1, \dotsc, a_n)$, we consider the family $\mathcal{M}_n(\mathbf{a})$ of $n \times n$…

The present work investigates regular, semiregular, and chiral polytopes of any rank $d\geq 3$, whose automorphism groups are 2-groups. There is a large variety of rather small finite regular or alternating semiregular polytopes with…

There is a longstanding conjecture, due to Gregory Cherlin and Boris Zilber, that all simple groups of finite Morley rank are simple algebraic groups. One of the major theorems in the area is Borovik's trichotomy theorem. The "trichotomy"…

We study asymmetric regular types. If $\frak p$ is regular and $A$-asymmetric then there exists a strict order such that Morley sequences in $\frak p$ over $A$ are strictly increasing (we allow Morley sequences to be indexed by elements of…

We construct two infinite series of Moufang loops of exponent $3$ whose commutative center (i.e. the set of elements that commute with all elements of the loop) is not a normal subloop. In particular, we obtain examples of such loops of…

Let $G$ be a permutation group on a set $\Omega$. A subset of $\Omega$ is a base for $G$ if its pointwise stabilizer in $G$ is trivial. By $b(G)$ we denote the size of the smallest base of $G$. Every permutation group with $b(G)=2$ contains…

The problem of classifying tuples of nilpotent matrices over a field under simultaneous conjugation is considered "hopeless". However, for any given matrix order over a finite field, the number of concerned orbits is always finite. This…

We show that for any positive integer $m\ge 1$, $m$-relator quotients of the modular group $M = PSL(2,\mathbb{Z})$ generically satisfy a very strong Mostow-type \emph{isomorphism rigidity}. We also prove that such quotients are generically…

The feeling that those two forcing notions-Cohen and Random-(equivalently the corresponding Boolean algebras Borel(R)/(meager sets), Borel(R)/(null sets)) are special, was probably old and widespread. A reasonable interpretation is to show…

We construct a CAT(0) group containing a finitely presented subgroup with infinitely many conjugacy classes of finite-order elements. Unlike previous examples (which were based on right-angled Artin groups) our ambient CAT(0) group does not…

All known Moufang sets arise, in some way or another, from an algebraic structure which can be called `division' in some way. In this PhD dissertation, I made an attempt to develop a theory of local Moufang sets, which generalize Moufang…

An abstract group $G$ is called totally $2$-closed if $H=H^{(2),\Omega}$ for any set $\Omega$ with $G\cong H\leq{\rm Sym}(\Omega)$, where $H^{(2),\Omega}$ is the largest subgroup of ${\rm Sym}(\Omega)$ whose orbits on $\Omega\times\Omega$…

The homological properties of localizations and completions of metabelian groups are studied. It is shown that, for $R=\mathbb Q$ or $R=\mathbb Z/n$ and a finitely presented metabelian group $G$, the natural map from $G$ to its…

We show that there is a pair of homogeneous polynomials such that the sets of roots of their Bernstein-Sato polynomials which are strictly supported at the origin are different although the sets of roots which are not strictly supported at…

We show that generic automorphisms of stable groups are supertight in a strong sense. In particular, we obtain the existence of supertight automorphisms. We also answer a question concerning the relationship between supertight automorphisms…

The rank of a hierarchically hyperbolic space is the maximal number of unbounded factors in a standard product region. For hierarchically hyperbolic groups, this coincides with the maximal dimension of a quasiflat. Examples for which the…

We show that the geometry associated to certain non-split sharply 2-transitive groups does not contain a proper projective plane. For a sharply 2-transitive group of finite Morley rank we improve known rank inequalities for this geometry…

It is known that every torsion-free abelian group of finite rank has a maximal completely decomposable summand that is unique up to isomorphism. We show that groups of infinite rank need not have maximal completely decomposable summands,…

The rational cohomology of the moduli space of rank two, odd degree stable bundles over a curve (of genus g > 1) has been studied intensely in recent years and in particular the invariant subring generated by Newstead's generators alpha,…

We construct finitely generated torsion-free solvable groups $G$ that have infinite rank, but such that all finitely generated torsion-free metabelian subquotients of $G$ are virtually abelian. In particular all finitely generated…