Related papers: HR-length of a free group via polynomial functors

We construct an action of the free group $F_n$ on the homotopy category of projective modules over a finite dimensional zigzag algebra. The main theorem in the paper is that this action is faithful. We describe the relationship between…

In this note we prove injectivity and relative asphericity for "layered" systems of equations over torsion-free groups, when the exponent matrix is invertible over Z. We also give elementary geometric proofs of results due to Bogley-Pride…

We establish upper bounds on the size of the largest subset of $\{1,2,\dots,N\}$ lacking nonzero differences of the form $h(p_1,\dots,p_{\ell})$, where $h\in \mathbb{Z}[x_1,\dots,x_{\ell}]$ is a fixed polynomial satisfying appropriate…

We show that every $H$-minor-free graph that also excludes a $(k \times k)$-grid as a minor has treewidth/branchwidth bounded from above by a function $f(t,k)$ that is linear in $k$ and polynomial in $t := |V(H)|$. Such a result was proven…

We study in this article stable homology of automorphism groups of free groups with coefficients twisted by a poynomial functor. We show that this homology is zero for a reduced covariant polynomial functor. For a reduced contravariant…

We study the growth of the order of torsion subgroups of the homology in a tower of finite abelian coverings. In particular, we prove that it is exponential for when the tower converges to the maximal free abelian cover of a link complement…

We prove that a homomorphism between free groups of finite rank equipped with the bi-invariant word metrics is a quasi-isometry if and only if it is an isomorphism.

According to the Furstenberg-Zimmer structure theorem, every measure-preserving system has a maximal distal factor, and is weak mixing relative to that factor. Furstenberg and Katznelson used this structural analysis of measure-preserving…

The $p$-group generation algorithm is used to verify that the Hilbert $3$-class field tower has length $3$ for certain imaginary quadratic fields $K$ with $3$-class group $\mathrm{Cl}_3(K) \cong [3,3]$. Our results provide the first…

A function $f: \mathbb{Z} \to \mathbb{Q}^n$ is a $c$-quasihomomorphism if the Hamming distance between $f(x+y)$ and $f(x)+f(y)$ is at most $c$ for all $x,y \in \mathbb{Z}$. We show that any $c$-quasihomomorphism has distance at most some…

Let F be a local field of positive characteristic, and let G be either a Heisenberg group over F, or a certain (nonabelian) two-dimensional unipotent group over F. If H is an arithmetic subgroup of G, we provide an explicit description of…

Firstly, we prove that every closed subgroup $H$ of type-preserving automorphisms of a locally finite thick affine building $\Delta$ of dimension $\geq 2$ that acts strongly transitively on $\Delta$ is Moufang. If moreover $\Delta$ is…

Let $K$ be a field of characteristic $p>0$, $A=K[[Y]]$ be a power series ring in one variable and $Q(A)$ be the field of fraction of $A$. Suppose that $R=A[X_1,\ldots,X_n]$ is a standard $\mathbb{N}^n$-graded polynomial ring over $A$, i.e.,…

Using the functor of Baumslag rationalization of groups we construct a functor on the category of all (non necessarily simply connected) spaces that extends the classical rationalization of simply connected spaces. We study this functor and…

Terao's factorization theorem shows that if an arrangement is free, then its characteristic polynomial factors into the product of linear polynomials over the integer ring. This is not a necessary condition, but there are not so many…

A discrete group G has periodic cohomology over R if there is an element in a cohomology group, cup product with which induces an isomorphism in cohomology after a certain dimension. Adem and Smith showed if R = Z, then this condition is…

This paper proves a criterion for verifying the monoid axiom in enriched left Bousfield localizations.

Let X be a homogeneous tree of degree q+1 (for q between 2 and infinity) and let f be a complex function on X times X for which f(x,y) only depend on the distance between x and y in X. Our main result gives a necessary and sufficient…

We give an example of a subgroup of SL(2,C) which is a strictly ascending HNN extension of a non-abelian finitely generated free group F. In particular, we exhibit a free group F in SL(2,C) of rank 6 which is conjugate to a proper subgroup…

For every $\alpha \in [1,2r-1]$, we show there exists a subgroup $H<F_r$ whose growth rate is $\alpha$.