Related papers: On definable $f$-generic groups and minimal flows …

Let $\mathbb K=(K,+,\cdot,v,\Gamma)$ be a valued algebraically closed field of characteristic and $(G,\oplus)$ be a $\mathcal K$-interpretable group that is either locally isomorphic to $(K,+)$ or to $(K,\cdot)$. Then if $\mathcal…

Let $W$ be a vector space over an algebraically closed field $k$. Let $H$ be a quasisimple group of Lie type of characteristic $p\ne {\rm char}(k)$ acting irreducibly on $W$. Suppose also that $G$ is a classical group with natural module…

Given a discrete quantum group $H$ with a finite normal quantum subgroup $G$, we show that any positive, possibly unbounded, harmonic function on $H$ with respect to an irreducible invariant random walk is $G$-invariant. This implies that,…

Let $G=G(K)$ be a simple algebraic group defined over an algebraically closed field $K$ of characteristic $p>0$. A subgroup $X$ of $G$ is said to be $G$-completely reducible if, whenever it is contained in a parabolic subgroup of $G$, it is…

Let $L(G)$ denote the space of integer-valued length functions on a countable group $G$ endowed with the topology of pointwise convergence. Assuming that $G$ does not satisfy any non-trivial mixed identity, we prove that a generic (in the…

Assume $G$ is a definable group in a stable structure $M$. Newelski showed that the semigroup $S_G(M)$ of complete types concentrated on $G$ is an inverse limit of the $\infty$-definable (in $M^{eq}$) semigroups $S_{G,\Delta}(M)$. He also…

We introduce two new classes of fusion categories which are obtained by a certain procedure from finite groups - weakly group-theoretical categories and solvable categories. These are fusion categories that are Morita equivalent to iterated…

Suppose $G$ is a finite group and $p$ is either a prime number or $0$. For $p$ positive, we say that $G$ is weakly tame at $p$ if $G$ has no non-trivial normal $p$-subgroups. By convention we say that every finite group is weakly tame at…

In this paper, using definability of types over indiscernible sequences as a template, we study a property of formulas and theories called "uniform definability of types over finite sets" (UDTFS). We explore UDTFS and show how it relates to…

We continue investigating the structure of externally definable sets in NIP theories and preservation of NIP after expanding by new predicates. Most importantly: types over finite sets are uniformly definable; over a model, a family of…

Consider the (formal/analytic/algebraic) map-germs Maps(X,(k^p,o)). Let G be the group of right/contact/left-right transformations. I extend the following (classical) results from the real/complex-analytic case to the case of arbitrary…

Motivated by the notion of strong computable type for sets in computable analysis, we define the notion of strong computable type for $G$-shifts, where $G$ is a finitely generated group with decidable word problem. A $G$-shift has strong…

For any subset $Z \subseteq \mathbb{Q}$, consider the set $S_Z$ of subfields $L\subseteq \overline{\mathbb{Q}}$ which contain a co-infinite subset $C \subseteq L$ that is universally definable in $L$ such that $C \cap \mathbb{Q}=Z$. Placing…

We prove that many properties and invariants of definable groups in NIP theories, such as definable amenability, G/G^{00}, etc., are preserved when passing to the theory of the Shelah expansion by externally definable sets, M^{ext}, of a…

Let $H$ be a subgroup of a finite group $G$. We say that $H$ satisfies partial $\Pi$-property in $G$ if there exists a chief series $\mathit{\Gamma}_G:1=G_0<G_1<\cdots<G_n=G$ of $G$ such that for every $G$-chief factor $G_i/G_{i-1}$ ($1\leq…

Let $p$ be a prime number. We consider diagonal $p$-permutation functors over a (commutative, unital) ring $\mathsf{R}$ in which all prime numbers different from $p$ are invertible. We first determine the finite groups $G$ for which the…

There are strong analogies between groups definable in o-minimal structures and real Lie groups. Nevertheless, unlike the real case, not every definable group has maximal definably compact subgroups. We study definable groups G which are…

We obtain many results and solve some problems about feebly compact paratopological groups. We obtain necessary and sufficient conditions for such a group to be topological. One of them is the quasiregularity. We prove that each…

We prove that infinite definably simple locally finite groups of finite centraliser dimension are simple groups of Lie type over locally finite fields. Then, we identify conditions on automorphisms of a stable group that make it resemble…

A graph polynomial $P$ is weakly distinguishing if for almost all finite graphs $G$ there is a finite graph $H$ that is not isomorphic to $G$ with $P(G)=P(H)$. It is weakly distinguishing on a graph property $\mathcal{C}$ if for almost all…