Related papers: The equalizer conjecture for the free group of ran…

Let f: G=* G(i) -> B=* B(i) be a group homomorphism between free products of groups. Suppose that G(i)f=B(i) of all i. Let H be a subgroup of G such that Hf=B. Then H decomposes into a free product H=*H(i) with H(i)f=B(i). Furthermore, H(i)…

Let $A$ and $B$ be unital separable simple amenable \CA s which satisfy the Universal Coefficient Theorem. Suppose {that} $A$ and $B$ are $\mathcal Z$-stable and are of rationally tracial rank no more than one. We prove the following:…

This paper concerns rigidity of the mapping class groups. We show that any homomorphism $\phi:{\rm Mod}_g\to {\rm Mod}_h$ between mapping class groups of closed orientable surfaces with distinct genera $g>h$ is trivial if $g\geq 3$ and has…

Let $G$ be a finite group acting effectively on the complex affine plane. If the $G$-action commutes with an \'etale endomorphism $f$ of the affine plane and the order of $G$ is even then the endomorphism $f$ is an automorphism.

Let $A$ be a separable $C^*$-algebra and let $B$ be a stable $C^*$-algebra with a strictly positive element. We consider the (semi)group $\Ext^{as}(A,B)$ (resp. $\Ext(A,B)$) of homotopy classes of asymptotic (resp. of genuine) homomorphisms…

Let $G$ be a metric group and let $\sA ut(G)$ denote the automorphism group of $G$. If $\sA$ and $\sB$ are groups of $G$-valued maps defined on the sets $X$ and $Y$, respectively, we say that $\sA$ and $\sB$ are \emph{equivalent} if there…

We prove that the comparison map from $G$-fixed points to $G$-homotopy fixed points, for the $G$-fold smash power of a bounded below spectrum $B$, becomes an equivalence after $p$-completion if $G$ is a finite $p$-group and $H_*(B; F_p)$ is…

Let $T_P$ be the theory of beautiful pairs of algebraically closed fields of fixed characteristic. It is known that for real tuples in models of $T_P$, SU-rank coincides with Morley rank and can be computed effectively. Building on Pillay's…

In this paper, we are motivated by two conjectures proposed by C. Bender et al.\ in 2024, which have remained open questions. The first conjecture states that if the complemented zero-divisor graph \( G(S) \) of a commutative semigroup \( S…

We prove that given a finite rank free group $\mathbb{F}$ of rank $\geq 3$ and two exponentially growing outer automorphisms $\psi$ and $\phi$ with dual lamination pairs $\Lambda^\pm_\psi$ and $\Lambda^\pm_\phi$ associated to them, and…

Let $A$ be an $n$-dimensional algebra over a field $k$ and $a(A)$ its quantum symmetry semigroup. We prove that the automorphisms group ${\rm Aut}_{\rm Alg} (A)$ of $A$ is isomorphic to the group $U \bigl( G(a (A)^{\rm o} ) \bigl)$ of all…

Let $U$ and $V$ be finite-dimensional vector spaces over a (commutative) field $\mathbb{K}$, and $\mathcal{S}$ be a linear subspace of the space $\mathcal{L}(U,V)$ of all linear operators from $U$ to $V$. A map $F : \mathcal{S} \rightarrow…

We first develop a general theory of Johnson filtrations and Johnson homomorphisms for a group $G$ acting on another group $K$ equipped with a filtration indexed by a "good" ordered commutative monoid. Then, specializing it to the case…

Stallings folding theory is modified, using double coset representatives, and to applied to the study of subgroups of amalgamated products of finite rank free groups. As a first application the subgroup membership problem for such groups is…

For two semi-simple algebras $A$ and $B$ over an arbitrary ground field $F$, we give a numerical criterion when $\Hom_F(A,B)$, the set of $F$-algebra homomorphisms between them, is non-empty. We also determine when the orbit set $B^\times…

A matching $M$ in a graph $G$ is {\em connected} if $G$ has an edge linking each pair of edges in $M$. The problem to find large connected matchings in graphs $G$ with $\alpha(G)=2$ is closely related to Hadwiger's conjecture for graphs…

Let $F$ be a finitely generated free group and let $H\le F$ be a finitely generated subgroup. Given an element $g\in F$, we study the ideal $\mathfrak{I}_g$ of equations for $g$ with coefficients in $H$, i.e. the elements $w(x)\in H*\langle…

The holomorphy conjecture predicts that the local Igusa zeta function associated to a hypersurface and a character is holomorphic on $\mathbb{C}$ whenever the order of the character does not divide the order of any eigenvalue of the local…

We study subgroups $H_U$ of the R. Thompson group $F$ which are stabilizers of finite sets $U$ of numbers in the interval $(0,1)$. We describe the algebraic structure of $H_U$ and prove that the stabilizer $H_U$ is finitely generated if and…

Let F_n be a free group of rank n>1. Two elements g, h in F_n are said to be translation equivalent in F_n if the cyclic length of \phi(g) equals the cyclic length of \phi(h) for every automorphism \phi of F_n. Let F(a, b) be the free group…