Related papers: Group Width
We extend the definitions of complexity measures of functions to domains such as the symmetric group. The complexity measures we consider include degree, approximate degree, decision tree complexity, sensitivity, block sensitivity, and a…
Except for crystalline or random structures, an agreed definition of complexity for intermediate and hence interesting cases does not exist. We fill this gap with a notion of complexity that characterises shapes formed by any finite number…
The Weisfeiler-Leman (WL) algorithms form a family of incomplete approaches to the graph isomorphism problem. They recently found various applications in algorithmic group theory and machine learning. In fact, the algorithms form a…
We study fundamental groups of non compact Riemannian manifolds. We find conditions which ensure that the fundamental group is trivial, finite or finitely generated.
We study how small is the set of critical values of the distance function from a compact (resp. closed) set in the plane or in a connected complete two-dimensional Riemannian manifold. We show that for a compact set, the set of critical…
Cieliebak, Mundet i Riera and Salamon recently formulated a definition of branched submanifold of Euclidean space in connection with their discussion of multivalued sections and the Euler class. This note proposes an intrinsic definition of…
For an analytic and univalent function $f$ in the unit disk $\mathbb{D}:=\{z\in\mathbb{C}:|z|<1\}$ with the normalization $f(0)=0=f'(0)-1$, the logarithmic coefficients $\gamma_n$ are defined by $\log \frac{f(z)}{z}= 2\sum_{n=1}^{\infty}…
In a family of compact, canonically polarized, complex manifolds the first variation of the lengths of closed geodesics is computed. As an application, we show the coincidence of the Fenchel-Nielsen and Weil-Petersson symplectic forms on…
For a finite word $w$ we define and study the Kolmogorov structure function $h_w$ for nondeterministic automatic complexity. We prove upper bounds on $h_w$ that appear to be quite sharp, based on numerical evidence.
Our goal is to develop a limit approach for a class of problems in additive combinatorics that is analogous to the limit theory of dense graph sequences. We introduce metric, convergence and limit objects for functions on groups and for…
The equational complexity function $\beta_\mathscr{V}:\mathbb{N}\to\mathbb{N}$ of an equational class of algebras $\mathscr{V}$ bounds the size of equation required to determine membership of $n$-element algebras in $\mathscr{V}$. Known…
We describe an effective version of the conjugacy problem and study it for wreath products and free solvable groups. The problem involves estimating the length of short conjugators between two elements of the group, a notion which leads to…
We introduce and study the concept of cyclicity degree of a finite group $G$. This quantity measures the probability of a random subgroup of $G$ to be cyclic. Explicit formulas are obtained for some particular classes of finite groups. An…
For a finite Coxeter group $W$ and $w$ an element of $W$ the `excess' of $w$ is defined to be $e(w) = \min\{\ell(x) + \ell(y) - \ell(w) \; | \; w=xy, \; x^2 = y^2 = 1\}$ where $\ell$ is the length function on $W$. Here we investigate the…
In this monograph, we give an account of the relationship between the algebraic structure of finitely generated and countable groups and the regularity with which they act on manifolds. We concentrate on the case of one--dimensional…
The concept of convex compactness, weaker than the classical notion of compactness, is introduced and discussed. It is shown that a large class of convex subsets of topological vector spaces shares this property and that is can be used in…
Several results are obtained concerning multiplicities of zeros of the Riemann zeta-function $\zeta(s)$. They include upper bounds for multiplicities, showing that zeros with large multiplicities have to lie to the left of the line $\sigma…
Irreducible sigma models, i.e. those for which the partition function does not factorise, are defined on Riemannian spaces with irreducible holonomy groups. These special geometries are characterised by the existence of covariantly constant…
The following two assertions are equivalent for an o-minimal expansion of an ordered group $\mathcal M=(M,<,+,0,\ldots)$. There exists a definable bijection between a bounded interval and an unbounded interval. Any definable continuous…
A function group is a finitely generated Kleinian group with an invariant connected component of its region of discontinuity. An extended function group is a finitely generated extended Kleinian group that contains orientation reversing…