Related papers: Nearly Maximal Hausdorff Dimension in Finitely Con…

Let G be a finite group of order n and V an irreducible representation over the complex numbers of dimension d. For some nonnegative number e, we have n=d(d+e). If e is small, then the character of V has unusually large degree. We fix e and…

We show that the kernel of the monodromy representation for hypersurfaces of degree d and dimension n is large for d at least three with the exception of the cases (d,n) = (3,0) and (3,1). For these the kernel is finite. By "large" we mean…

Many well-studied problems in extremal combinatorics deal with the maximum possible size of a family of objects in which every pair of objects satisfies a given restriction. One problem of this type was recently raised by Alon, Gujgiczer,…

We study a model of random $\mathcal{R}$-enriched trees that is based on weights on the $\mathcal{R}$-structures and allows for a unified treatment of a large family of random discrete structures. We establish distributional limits…

For a wide class of groups including polycyclic and finitely generated polynomial growth groups it is proved that the Reidemeister number of an automorphism f is equal to the number of finite-dimensional fixed points of the induced map f^…

We construct finitely generated simple torsion-free groups with strong homological control. Our main result is that every subset of $\mathbb{N} \cup \{\infty\}$, with some obvious exceptions, can be realized as the set of dimensions of…

The set of badly approximable $m \times n $ matrices is known to have Hausdorff dimension $mn $. Each such matrix comes with its own approximation constant $c$, and one can ask for the dimension of the set of badly approximable matrices…

For any conformal iterated function system (CIFS) consisting of finitely or countably many maps, and any closed shift-invariant set of right-infinite sequences of such maps, one can associate a limit set, which we call a shift-generated…

We study random bipartite planar maps defined by assigning nonnegative weights to each face of a map. We prove that for certain choices of weights a unique large face, having degree proportional to the total number of edges in the maps,…

We prove partial regularity for minimizers to elasticity type energies in the nonlinear framework {with $p$-growth, $p>1$,} in dimension $n\geq 3$. It is an open problem in such a setting either to establish full regularity or to provide…

I compute the structure of the restricted 2-algebra associated to a group first described by Andrew Brunner, Said Sidki and Ana Cristina Vieira, acting on the binary rooted tree. I show that its width is unbounded, growing logarithmically,…

We study the finiteness properties of the braided Higman--Thompson group $bV_{d,r}(H)$ with labels in $H\leq B_d$, and $bF_{d,r}(H)$ and $bT_{d,r}(H)$ with labels in $H\leq PB_d$ where $B_d$ is the braid group with $d$ strings and $PB_d$ is…

We give upper and lower bounds for the Hausdorff dimensions for a class of graph-directed measures when its underlying directed graph is the infinite N-ary tree. These measures are different from graph-directed self-similar measures driven…

The asymptotic dimension of metric spaces is an important notion in geometric group theory introduced by Gromov. The metric spaces considered in this paper are the ones whose underlying spaces are the vertex-sets of graphs and whose metrics…

For planar self-affine sets satisfying the strong separation condition, recent work of B\'ar\'any, Hochman, and Rapaport gives mild assumptions under which the Hausdorff dimension equals the affinity dimension. In this paper, we study…

In comparison to graphs, combinatorial methods for the isomorphism problem of finite groups are less developed than algebraic ones. To be able to investigate the descriptive complexity of finite groups and the group isomorphism problem, we…

We calculate the almost sure Hausdorff dimension of the random covering set $\limsup_{n\to\infty}(g_n + \xi_n)$ in $d$-dimensional torus $\mathbb T^d$, where the sets $g_n\subset\mathbb T^d$ are parallelepipeds, or more generally, linear…

We prove that, for any finite set of minimal $r$-graph patterns, there is a finite family $\mathcal F$ of forbidden $r$-graphs such that the extremal Tur\'an constructions for $\mathcal F$ are precisely the maximum $r$-graphs obtainable…

We prove that finite groups have the same complex character tables iff the group algebras are twisted forms of each other as Drinfel'd quasi-bialgebras or iff there is non-associative bi-Galois algebra over these groups. The interpretations…

For bounded pseudoconvex domains with finite type we give a precise description of the automorphism group: if an orbit of the automorphism group accumulates on at least two different points of the boundary, then the automorphism group has…