Related papers: Subobjects of the successive power objects in the …

We prove that the Higman-Thompson groups $G_{n,r}^+$ and $G_{m,s}^+$ are isomorphic if and only if $m=n$ and $\mbox{gcd}(n-1,r)=\mbox{gcd}(n-1,s)$.

We first show that every group-theoretical category is graded by a certain double coset ring. As a consequence, we obtain a necessary and sufficient condition for a group-theoretical category to be nilpotent. We then give an explicit…

The \emph{power graph} $P(G)$ of a group $G$ is the graph whose vertex set is $G$, with $x$ and $y$ joined if one is a power of the other; the \emph{directed power graph} $\vec{P}(G)$ has the same vertex set, with an arc from $x$ to $y$ if…

We study Subgraph Isomorphism on graph classes defined by a fixed forbidden graph. Although there are several ways for forbidding a graph, we observe that it is reasonable to focus on the minor relation since other well-known relations lead…

A P-set of a symmetric matrix $A$ is a set $\alpha$ of indices such that the nullity of the matrix obtained from $A$ by removing rows and columns indexed by $\alpha$ is $|\alpha|$ more than that of $A$. It is known that each subset of a…

Let $G$ be a complex reductive group and $V$ a $G$-module. Then the $m$th jet scheme $G_m$ acts on the $m$th jet scheme $V_m$ for all $m\geq 0$. We are interested in the invariant ring $\mathcal{O}(V_m)^{G_m}$ and whether the map…

We prove that the category of faded cosheaves in Set over a sober topological space $(B,\Omega)$ is equivalent to a category Sett$(B,\Omega)$ having the same class of objects as Set$ / B$ has, but generally a wider class of morphisms. We…

A topological space is reversible if each continuous bijection of it onto itself is open. We introduce an analogue of this notion in the category of topological groups: A topological group G is g-reversible if every continuous automorphism…

Let S be a subsemigroup of an abelian torsion-free group G. If S is a positive cone of G, then all C*-algebras generated by faithful isometrical non-unitary representations of S are canonically isomorphic. Proved by Murphy, this statement…

The independence polynomial $I(G, x)$ of a graph $G$ is the polynomial in variable $x$ in which the coefficient $a_n$ on $x^n$ gives the number of independent subsets $S \subseteq V(G)$ of vertices of $G$ such that $|S| = n$. $I(G, x)$ is…

We show that if $M$ is a sub-Riemannian manifold and $N$ is a Carnot group such that the nilpotentization of $M$ at almost every point is isomorphic to $N$, then there are subsets of $N$ of positive measure that embed into $M$ by…

The deck of a topological space $X$ is the set $\mathcal{D}(X)=\{[X \setminus \{x\}] \colon x \in X\}$, where $[Z]$ denotes the homeomorphism class of $Z$. A space $X$ is topologically reconstructible if whenever…

Let $X$ be a proper algebraic scheme over an algebraically closed field. We assume that a torus $T$ acts on $X$ such that the action has isolated fixed points. The $T$-graph of $X$ can be defined using the fixed points and the one…

Let $G$ be a locally compact group with cocompact connected component. We prove that the assembly map from the topological $\k$-theory of $G$ to the $\k$-theory of the reduced $C^*$-algebra of $G$ is an isomorphism.

The holomorph of a discrete group $G$ is the universal semi-direct product of $G$. In chapter 1 we describe why it is an interesting object and state main results. In chapter 2 we recall the classical definition of the holomorph as well as…

The consistent systems of idempotents of Meyer and Solleveld allow to construct Serre subcategories of $Rep_R(G)$, the category of smooth representations of a $p$-adic group $G$ with coefficients in $R$. In particular, they were used to…

Consider a reflection from a finitely-complete category $\mathbb{C}$ into its full subcategory $\mathbb{M}$, with unit $\eta :1_\mathbb{C}\rightarrow HI$. Suppose there is a left-exact functor $U$ into the category of sets, such that $UH$…

We give a necessary and sufficient graph-theoretic characterization of toric ideals of graphs that are unimodular. As a direct consequence, we provide the structure of unimodular graphs by proving that the incidence matrix of a graph $G$ is…

A matching set $M$ in a graph $G$ is a collection of edges of $G$ such that no two edges from $M$ share a vertex. In this paper we consider some parameters related to the matching of regular graphs. We find the sixth coefficient of the…

Let $\mathcal{U}$ be the unipotent variety of a complex reductive group $G$. Fix opposed Borel subgroups $B_\pm \subseteq G$ with unipotent radicals $U_\pm$. The map that sends $x_+x_- \mapsto x_+x_-x_+^{-1}$ for all $x_\pm \in U_\pm$…