Related papers: Basic and degenerate pregeometries

We study finite four-valent graphs Gamma admitting an edge-transitive group G of automorphisms such that G determines and preserves an edge-orientation on Gamma, and such that at least one G-normal quotient is a cycle (a quotient modulo the…

We construct relative Gromov--Witten theory with expanded degenerations in the normal crossings setting and establish a degeneration formula for the resulting invariants. Given a simple normal crossings pair $(X,D)$, we show that there…

Let $G/\Gamma$ be the quotient of a semisimple Lie group by an arithmetic lattice. We show that for reductive subgroups $H$ of $G$ that is large enough, the orbits of $H$ on $G/\Gamma$ intersect nontrivially with a fixed compact set. As a…

Let G:=SO(n,1)^\circ and \Gamma be a geometrically finite Zariski dense subgroup with critical exponent delta bigger than (n-1)/2. Under a spectral gap hypothesis on L^2(\Gamma \ G), which is always satisfied for delta>(n-1)/2 for n=2,3 and…

A space of pseudoquotients $\mathcal{B}(X,S)$ is defined as equivalence classes of pairs $(x,f)$, where $x$ is an element of a non-empty set $X$, $f$ is an element of $S$, a commutative semigroup of injective maps from $X$ to $X$, and…

In this paper we discuss a method for bounding the size of the stabiliser of a vertex in a $G$-vertex-transitive graph $\Gamma$. In the main result the group $G$ is quasiprimitive or biquasiprimitive on the vertices of $\Gamma$, and we…

Consider a locally compact quantum group $\mathbb{G}$ with a closed classical abelian subgroup $\Gamma$ equipped with a $2$-cocycle $\Psi:\hat{\Gamma}\times\hat{\Gamma}\to\mathbb{C}$. We study in detail the associated Rieffel deformation…

Let $\C(\Gamma)$ be the set of isomorphism classes of the finite groups that are homomorphic images of $\Gamma$. We investigate the extent to which $\C(\Gamma)$ determines $\Gamma$ when $\Gamma$ is a group of geometric interest. If…

Let PG$(\mathbb{F}_q^v)$ be the $(v-1)$-dimensional projective space over $\mathbb{F}_q$ and let $\Gamma$ be a simple graph of order ${q^k-1\over q-1}$ for some $k$. A 2$-(v,\Gamma,\lambda)$ design over $\mathbb{F}_q$ is a collection $\cal…

To each multiquiver $\Gamma$ we attach a solution to the consistency equations associated to twisted generalized Weyl (TGW) algebras. This generalizes several previously obtained solutions in the literature. We show that the corresponding…

A regular bipartite graph $\Gamma$ is called semisymmetric if its full automorphism group $\mathrm{Aut}(\Gamma)$ acts transitively on the edge set but not on the vertex set. For a subgroup $G$ of $\mathrm{Aut}(\Gamma)$ that stabilizes the…

Diagonal groups are one of the classes of finite primitive permutation groups occurring in the conclusion of the O'Nan-Scott theorem. Several of the other classes have been described as the automorphism groups of geometric or combinatorial…

Let $H$ be a normal subgroup of a group $G$. The normal subgroup based power graph $\Gamma_H(G)$ of $G$ is the simple undirected graph with vertex set $V(\Gamma_H(G))= (G\setminus H)\cup \{e\}$ and two distinct vertices $a$ and $b$ are…

Given a strict partial order $\Delta$ on a set $\Lambda$ and an arbitrary ring $R$ with $1\neq 0$, the corresponding McLain group $M(\Delta)$ has been studied in depth. We construct a larger family of McLain groups $G(\Delta)$, where…

Let $M$ be complete flat pseudo-Riemannian homogeneous manifold and $\Gamma\subset\Iso(\RR^n_s)$ its fundamental group. We show that $M$ is a trivial fiber bundle $G/\Gamma\to M\to\RR^{n-k}$, where $G$ is the Zariski closure of $\Gamma$ in…

Let $G$ be a nontrivial transitive permutation group on a finite set $\Omega$. An element of $G$ is said to be a derangement if it has no fixed points on $\Omega$. From the orbit counting lemma, it follows that $G$ contains a derangement,…

Given a finite group $G$ with a normal subgroup $N$, the simple graph $\Gamma_\textit{G}( \textit{N} )$ is a graph whose vertices are of the form $|x^G|$, where $x\in{N\setminus{Z(G)}}$, and $x^G$ is the $G$-conjugacy class of $N$…

Let $G$ be a transitive permutation group on a finite set $\Omega$ and recall that a base for $G$ is a subset of $\Omega$ with trivial pointwise stabiliser. The base size of $G$, denoted $b(G)$, is the minimal size of a base. If $b(G)=2$…

We analyse the normal quotient structure of several infinite families of finite connected edge-transitive, four-valent oriented graphs. These families were singled out by Marusic and others to illustrate various different internal…

Let $A$ be a central simple algebra over a number field $K$ with ring of integers $\mathcal{O}_K$, such that either the degree of the algebra $n \ge 3$, or $n=2$ and $A$ is not a totally definite quaternion algebra. Then strong…