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Let k be a field, G a smooth connected linear algebraic group and X a homogeneous space of G over k, such that the geometric stabilizers are extensions of a smooth group of multiplicative type by a smooth connected characterfree group. If k…

Let G be a discrete group. We give methods to compute for a generalized (co-)homology theory its values on the Borel construction (EG x X)/G of a proper G-CW-complex X satisfying certain finiteness conditions. In particular we give formulas…

Let $ L/K $ be a finite separable extension of fields whose Galois closure $ E/K $ has group $ G $. Greither and Pareigis have used Galois descent to show that a Hopf algebra giving a Hopf-Galois structure on $ L/K $ has the form $ E[N]^{G}…

Let l be an odd prime and K/k a Galois extension of totally real number fields with Galois group G such that K/k_\infty and k/Q are finite. We give a full description of the algebraic structure of the semisimple algebra QG=Quot(\Lambda G)…

Let $H$ be an acylindrically hyperbolic group without nontrivial finite normal subgroups. We show that any finite system $S$ of equations with constants from $H$ is equivalent to a single equation. We also show that the algebraic set…

Given a quasi-reductive group $G$ over a local field $k$, using Berkovich geometry, we exhibit a family of $G(k)$-equivariant compactifications of the Bruhat-Tits building $\mathcal B(G, k)$, constructed and investigated by Solleveld and…

Given a finite category T, we consider the functor category [T,A], where A can in particular be any quasi-abelian category. Examples of quasi-abelian categories are given by any abelian category but also by non-exact additive categories as…

We consider compact homogeneous spaces G/H, where G is a compact connected Lie group and H is its closed connected subgroup of maximal rank. The aim of this paper is to provide an effective computation of the universal toric genus for the…

In this paper we obtain new quantitative forms of Hilbert's Irreducibility Theorem. In particular, we show that if $f(X, T_1, \ldots, T_s)$ is an irreducible polynomial with integer coefficients, having Galois group $G$ over the function…

Let G be a split semi-simple p-adic group and let H be its Iwahori-Hecke algebra with coefficients in the algebraic closure k of the finite field with p elements. Let F be the affine flag variety over k associated with G. We show, in the…

Let $L/K$ be a finite Galois extension whose Galois group $G$ is non-abelian and characteristically simple. Using tools from graph theory, we shall give a closed formula for the number of Hopf-Galois structures on $L/K$ with associated…

Let $\Ga$ be a connected, solvable linear algebraic group over a number field~$K$, let $S$ be a finite set of places of~$K$ that contains all the infinite places, and let $\theints$ be the ring of $S$-integers of~$K$. We define a certain…

We prove that if $G$ is a noncompact connected real reductive linear Lie group, then any discrete subgroup of $G$ acting properly discontinuously and cocompactly on some homogeneous space $G/H$ of $G$ is quasi-isometrically embedded and…

The description of irreducible representations of a group G can be seen as a question in harmonic analysis; namely, decomposing a suitable space of functions on G into irreducibles for the action of G x G by left and right multiplication.…

We give another proof that a reductive algebraic group is geometrically reductive. We show that a quotient of the semi-stable locus (by a linear action of a reductive algebraic group on a projective scheme) exists, and from this Haboush's…

Let k be a non archimedean field. If X is a k-algebraic variety and U a locally closed semi-algebraic subset of X^{an} -- the Berkovich space associated to X -- we show that for l \neq char(\tilde{k}), the cohomology groups H^i_c (\bar{U},…

To a "stable homotopy theory" (a presentable, symmetric monoidal stable $\infty$-category), we naturally associate a category of finite \'etale algebra objects and, using Grothendieck's categorical machine, a profinite group that we call…

Let $A$ be a complex torus and $G$ a finite group acting on $A$ without translations such that $A/G$ is smooth. Consider the subgroup $F\leq G$ generated by elements that have at least one fixed point. We prove that there exists a point…

The absolute Galois group Gal$(\overline{\mathbb{Q}}/\mathbb{Q})$ of the field $\mathbb{Q}$ of rational numbers can be presented as a highly computable object, under the notion of type-2 Turing computation. We formalize such a presentation…

Let $k_0$ be a field of characteristic $p>0$ and $k=k_0(t)$, where $t$ is transcendental over $k_0$. We give an example of a smooth connected unipotent $k$-group $G$ such that $G(F)/R$ is non-commutative for some finite separable field…