Related papers: Flexible stability and nonsoficity

Let G be a finitely generated linear group over a field of characteristic 0. Suppose that every solvable subgroup of G is polycyclic. Then the claim is made that any solvable subgroup of G is separable. This is proven for G=SL_n(Z).…

We show that a dynamical sequence $(f_n)_{n\in \mathbb{N}}$ of polynomials over a number field whose set of stable primes is of positive density must necessarily have a very restricted, and in particular ``near-solvable" dynamical Galois…

Given a graph $F$, a hypergraph is called a Berge-$F$ if it can be obtained by expanding each edge of $F$ into a hyperedge containing it. Let $M_{k}$ denote the matching of size $k$. Kang, Ni, and Shan [12] determined the Tur\'an number of…

A group is said to be cube-free if its order is not divisible by the cube of any prime. Let $f_{cf,sol}(n)$ denote the isomorphism classes of solvable cube-free groups of order $n$. We find asymptotic bounds for $f_{cf,sol}(n)$ in this…

For an algebraically closed field K, let G be a finite abelian group of K-linear automorphisms of a finite-dimensional path algebra KQ of a quiver Q. Under certain assumptions on the action of G, we show the existence of a certain kind of…

A graph $G$ is \emph{unstable} if its canonical double cover CDC$(G)$ has more automorphisms than Aut$(G)\times \mathbb{Z}_2$. A related problem asks when two non-isomorphic graphs share the same CDC. We unify both via \emph{lifting} and…

For a finite noncyclic group $G$, let $\Cyc(G)$ be a set of elements $a$ of $G$ such that $\langle a,b\rangle$ is cyclic for each $b$ of $G$. The noncyclic graph of $G$ is a graph with the vertex set $G\setminus \Cyc(G)$, having an edge…

A discrete group is matricially stable if every function from the group to a complex unitary group that is "almost multiplicative" in the point-operator norm topology is "close" to a genuine unitary representation. It follows from a recent…

We study $\epsilon$-representations of discrete groups by unitary operators on a Hilbert space. We define the notion of Ulam stability of a group which loosely means that finite-dimensional $\epsilon$-represendations are uniformly close to…

A theorem of Waldspurger states that the Fourier transform of a stable distribution on the Lie algebra of a simply-connected semisimple group $G$ over a p-adic field, is again stable. We generalize this theorem to representations whose…

We prove that group homology of the diffeomorphism group of $\#^g S^n \times S^n$ as a discrete group is independent of $g$ in a range, provided that $n>2$. This answers the high dimensional version of a question posed by Morita about…

For $d \geq 4$ and $p$ a sufficiently large prime, we construct a lattice $\Gamma \leq {\rm PSp}_{2d}(\mathbb Q_p),$ such that its universal central extension cannot be sofic if $\Gamma$ satisfies some weak form of stability in…

In this paper we investigate graphs that admit a group acting arc-transitively such that the local action is semiprimitive with a regular normal nilpotent subgroup. This type of semiprimitive group is a generalisation of an affine group. We…

Considering a finite group $G$, for any element $x\in G$, the solvabilizer of $x$ in $G$ is defined as $Sol_G(x)=\{y \in G : \langle x, y \rangle \text{ is solvable}\}$. In this paper, we introduce $Solv(G)$ as the number of distinct…

For all $k \ge 2$, we show that there exists a group $G$ and a non-free stably free $\mathbb{Z} G$-module of rank $k$. We use this to show that, for all $k \ge 2$, there exist homotopically distinct finite $2$-complexes with fundamental…

Let G be the Heisenberg group of real lower triangular 3x3 matrices with unit diagonal. A locally free smooth action of G on a manifold M^4 is given by linearly independent vector fields X_1, X_2, X_3 such that X_3 = [X_1,X_2] and [X_1,X_3]…

An edge-weighted graph $G=(V,E)$ is called stable if the value of a maximum-weight matching equals the value of a maximum-weight fractional matching. Stable graphs play an important role in some interesting game theory problems, such as…

We provide a quantitative formulation of the equivalence between hyperlinearity and soficity for amenable groups, showing that every hyperlinear approximation to such a group is essentially produced from a sofic approximation. This…

In order to be convergent, linear multistep methods must be zero stable. While constant step size theory was established in the 1950's, zero stability on nonuniform grids is less well understood. Here we investigate zero stability on…

Let G be a planar graph with a list assignment L. Suppose a preferred color is given for some of the vertices. We prove that if G has girth at least six and all lists have size at least three, then there exists an L-coloring respecting at…