Related papers: Rational fixed points for linear group actions

For a finitely generated free group F_n, of rank at least 2, any finite subgroup of Out(F_n) can be realized as a group of automorphisms of a graph with fundamental group F_n. This result, known as Out(F_n) realization, was proved by…

We give a criterion for group elements to have fixed points with respect to a semi-simple action on a complete CAT(0) space of finite topological dimension. As an application, we show that Thompson's group T and various generalizations of…

Let $G$ be a finite group, $k$ be a field and $G\to GL(V_{\rm reg})$ be the regular representation of $G$ over $k$. Then $G$ acts naturally on the rational function field $k(V_{\rm reg})$ by $k$-automorphisms. Define $k(G)$ to be the fixed…

Let $G$ be a compact connected Lie group, or more generally a path connected topological group of the homotopy type of a finite CW-complex, and let $X$ be a rational nilpotent $G$-space. In this paper we analyze the homotopy type of the…

Let $k$ be any field, $G$ be a finite group. Let $G$ act on the rational function field $k(x_g:g\in G)$ by $k$-automorphisms defined by $h\cdot x_g=x_{hg}$ for any $g,h\in G$. Denote by $k(G)=k(x_g:g\in G)^G$ the fixed field. Noether's…

Let $K$ be a field, $a, b\in K$ and $ab\neq 0$. Let us consider the polynomials $g_{1}(x)=x^n+ax+b, g_{2}(x)=x^n+ax^2+bx$, where $n$ is a fixed positive integer. In this paper we show that for each $k\geq 2$ the hypersurface given by the…

Let $G$ be a reductive affine algebraic group, and let $X$ be an affine algebraic $G$-variety. We establish a (poly)stability criterion for points $x\in X$ in terms of intrinsically defined closed subgroups $H_{x}$ of $G$, and relate it…

Given a $p$-adic group $G$ equipped with an action of a finite group $\Gamma\subset\mathrm{Aut}_F(\mathbf{G})$, and a reductive fixed-point subgroup $G^\Gamma$, we establish a relationship between constructions of types for these two groups…

Let $G$ be a simple algebraic group over an algebraically closed field and let $X$ be an irreducible subvariety of $G^r$ with $r \geqslant 2$. In this paper, we consider the general problem of determining if there exists a tuple $(x_1,…

We prove that the Farrell-Jones assembly map for connective algebraic K-theory is rationally injective, under mild homological finiteness conditions on the group and assuming that a weak version of the Leopoldt-Schneider conjecture holds…

Let $n$ be a positive integer and let $\mathbb F_{q^n}$ be the finite field with $q^n$ elements, where $q$ is a power of a prime. This paper introduces a natural action of the Projective Semilinear Group $\text{P}\Gamma \text{L}(2,…

A classical result by Solodov states that if a group acts on the line such that any non-trivial element has at most one fixed point, then the action is either abelian or semi-conjugate to an affine action. We show that the same holds if we…

Finite rank median spaces are a simultaneous generalisation of finite dimensional ${\rm CAT}(0)$ cube complexes and real trees. If $\Gamma$ is an irreducible lattice in a product of rank one simple Lie groups, we show that every action of…

This paper is a generalization of a previous paper by the author to connected unipotent linear algebraic groups. The notion of an $ \alpha $-pair answers when an open $ G $-stable, affine, sub-variety $ D(H) $ is a trivial bundle over $ G…

We prove new results on the distribution of rational points on ramified covers of abelian varieties over finitely generated fields $k$ of characteristic zero. For example, given a ramified cover $\pi : X \to A$, where $A$ is an abelian…

Let $M$ be an irreducible smooth projective variety, defined over an algebraically closed field, equipped with an action of a connected reductive affine algebraic group $G$, and let ${\mathcal L}$ be a $G$--equivariant very ample line…

Let G be a subgroup of GL(V), where V is a finite dimensional vector space over a finite field of characteristic p >0. If det(g-1) = 0 for all g \in G then we call G a fixed-point subgroup of GL(V). Motivated in parallel by questions in…

Let $X$ be a smooth algebraic variety endowed with an action of a finite group $G$ such that there exists the geometric quotient $\pi_X:X\to X/G$. We characterize rational tensor fields $\tau$ on $X/G$ such that the {\it pull back} of $\tau…

A discrete group which admits a faithful, finite dimensional, linear representation over a field $\mathbb F$ of characteristic zero is called linear. This note combines the natural structure of semi-direct products with work of A. Lubotzky…

We classify all rational maps $H \in K(x)^n$ for which ${\rm trdeg}_K K(tH_1,tH_2,\ldots,tH_n) \le 2$, where $K$ is any field and $t$ is another indeterminate. Furthermore, we classify all such maps for which additionally $JH \cdot H = {\rm…