Related papers: On $e$-local structures for $\mathbb{Z}_\ell$-spet…

In this paper we prove that various quasi-categories whose objects are $\infty$-categories in a very general sense are complete: admitting limits indexed by all simplicial sets. This result and others of a similar flavor follow from a…

These notes are defining the notion of centric linking system for a locally finite group If a locally finite group $G$ has countable Sylow $p$-subgroups, we prove that, with a countable condition on the set of intersections, the…

Let $q$ be a prime and $A$ an elementary abelian $q$-group acting as a coprime group of automorphisms on a profinite group $G$. We show that if $A$ is of order $q^2$ and some power of each element in $C_G(a)$ is Engel in $G$ for any $a\in…

Let G be a complex reductive Lie group acting on a compact K\"ahler manifold X and assume that the action of a maximal compact subgroup K of G is Hamiltonian. For each extreme point of the convex hull of the momentum map image, there is an…

For a smooth manifold $M$, possibly with boundary and corners, and a Lie group $G$, we consider a suitable description of gauge fields in terms of parallel transport, as groupoid homomorphisms from a certain path groupoid in $M$ to $G$.…

In this article, we study Sobolev homeomorphisms and composition operators on homogeneous Lie groups. We prove that a measurable homeomorphism $\varphi: \Omega \to\widetilde{\Omega}$ belongs to the Sobolev space $L^{1}_{q}(\Omega;…

For any topological bicategory B, the Duskin nerve NB of B is a simplicial space. We introduce the classifying topos BB of B as the Deligne topos of sheaves Sh(NB) on the simplicial space NB. It is shown that the category of geometric…

Consider a simple algebraic group $G$ of classical type and its Lie algebra $\mathfrak{g}$. Let $(e,h,f) \subset \mathfrak{g}$ be an $\mathfrak{sl}_2$-triple and $Q_e= C_G(e,h,f)$. The torus $T_e$ that comes from the…

Let $E$ be an optimal elliptic curve over $\Q$ of prime conductor $N$. We show that if for an odd prime $p$, the mod $p$ representation associated to $E$ is reducible (in particular, if $p$ divides the order of the torsion subgroup of…

Let G be a connected real reductive algebraic group, and let K be a maximal compact subgroup of G. We prove that the conjugation orbit space Hom(Z^{2d},K)/K is a strong deformation retract of the space Hom(Z^{2d},G)/G of equivalence classes…

Assume $G$ is a solvable group whose elementary abelian sections are all finite. Suppose, further, that $p$ is a prime such that $G$ fails to contain any subgroups isomorphic to $C_{p^\infty}$. We show that if $G$ is nilpotent, then the…

Let $G$ be a discrete group. We prove that the category of $G$-posets admits a model structure that is Quillen equivalent to the standard model structure on $G$-spaces. As is already true nonequivariantly, the three classes of maps defining…

Let G be a reductive group over a commutative ring R. We say that G has isotropic rank >=n, if every normal semisimple reductive R-subgroup of G contains (G_m)^n. We prove that if G has isotropic rank >=1 and R is a regular domain…

Let $V$ be a finite dimensional vector space equipped with a non-degenerate Hermitian form over a field $\mathbb{K}$. Let $\mathcal{G}(V)$ be the graph with vertex set the $1$-dimensional non-degenerate subspaces of $V$ and adjacency…

We develop the theory of reflective subfibrations on an $\infty$-topos $\mathcal{E}$. A reflective subfibration $L_\bullet$ on $\mathcal{E}$ is a pullback-compatible assignment of a reflective subcategory $\mathcal{D}_X\subseteq…

Let $\mc G$ be a reductive group over an algebraically closed field of characteristic $p>0$. We study homogeneous $\mc G$-spaces that are induced from the $G\times G$-space $G$, $G$ a suitable reductive group, along a parabolic subgroup of…

In this paper we describe two ways on which cofibred categories give rise to bisimplicial sets. The "fibred nerve" is a natural extension of Segal's classical nerve of a category, and it constitutes an alternative simplicial description of…

We show that for a differential graded Lie algebra $\mathfrak{g}$ whose components vanish in degrees below -1 the nerve of the Deligne 2-groupoid is homotopy equivalent to the simplicial set of $\mathfrak{g}$-valued differential forms…

We establish a purely geometric form of the concentration theorem (also called localization theorem) for actions of a linearly reductive group $G$ on an affine scheme $X$ over an affine base scheme $S$. It asserts the existence of a…

We study the homotopy properties of the posets of p-subgroups Sp(G) and Ap(G) of a finite group G, viewed as finite topological spaces. We answer a question raised by R.E. Stong in 1984 about the relationship between the contractibility of…