Related papers: Finding generators and relations for groups acting…
We study the problem of finding generators for the fundamental group G of a space of the following sort: one removes a family of complex hyperplanes from n dimensional complex vector space, or n dimensional complex hyperbolic space, or the…
We prove that every finitely generated group $G$ discriminated by a locally quasi-convex torsion-free hyperbolic group $\Gamma$ is effectively coherent: that is, presentations for finitely generated subgroups can be computed from the…
Networks representing many complex systems in nature and society share some common structural properties like heterogeneous degree distributions and strong clustering. Recent research on network geometry has shown that those real networks…
We give generators for a certain complex hyperbolic braid group. That is, we remove a hyperplane arrangement from complex hyperbolic $13$-space, take the quotient of the remaining space by a discrete group, and find generators for the…
We discover a simple construction of a four-dimensional family of smooth surfaces of general type with $p_g(S)=q(S)=0$, $K^2_S=3$ with cyclic fundamental group $C_{14}$. We use a degeneration of the surfaces in this family to find…
In the present paper we find generators of the mixed commutator subgroups of relative elementary groups and obtain unrelativised versions of commutator formulas in the setting of Bak's unitary groups. It is a direct sequel of our similar…
$ $Abert, Gelander and Nikolov [AGN17] conjectured that the number of generators $d(\Gamma)$ of a lattice $\Gamma$ in a high rank simple Lie group $H$ grows sub-linearly with $v = \mu(H / \Gamma)$, the co-volume of $\Gamma$ in $H$. We prove…
We give bounds on the degree of generators for the ideal of relations of the graded algebras of modular forms with coefficients in $\mathbb{Q}$ over congruence subgroups $\Gamma_0(N)$ for $N$ satisfying some congruence conditions and for…
This is an expository article about groups generated by two isometries of the complex hyperbolic plane.
Given a finite group $G$, the generating graph $\Gamma(G)$ of $G$ has as vertices the non-identity elements of $G$ and two vertices are adjacent if and only if they are distinct and generate $G$ as group elements. Let $G$ be a 2-generated…
Hypergraphs, which belong to the family of higher-order networks, are a natural and powerful choice for modeling group interactions in the real world. For example, when modeling collaboration networks, which may involve not just two but…
For any torsion-free hyperbolic group $\Gamma$ and any group $G$ that is fully residually $\Gamma$, we construct algorithmically a finite collection of homomorphisms from $G$ to groups obtained from $\Gamma$ by extensions of centralizers,…
Recently, Prasad and Yeung classified all possible fundamental groups of fake projective planes. According to their result, many fake projective planes admit a nontrivial group of automorphisms, and in that case it is isomorphic to…
Let $n$ be a prime or its square. We prove that the congruence subgroup $\Gamma_0(n)$ admits a free product decomposition into cyclic factors in such a way that the $(2,1)$-component of each cyclic generator is either $n$ or $0$, answering…
We show that the projectivized complex reflection group $\Gamma$ of the unique $(1+i)$-modular Hermitian $\mathbb{Z}[i]$-module of signature $(9,1)$ is a new arithmetic reflection group in $PU(9,1)$. We find $32$ complex reflections of…
By means of identical cubic elements, we generate a partition of a volume in which a particle-based cosmological simulation is carried out. In each cubic element, we determine the gas particles with a normalized density greater than an…
We study graded rings of modular forms over congruence subgroups, with coefficients in a subring $A$ of $\mathbb{C}$, and specifically the highest weight needed to generate these rings as $A$-algebras. In particular, we determine upper…
Given a discrete subgroup $\Gamma$ of finite co-volume of $\mathrm{PGL}(2,\mathbb{R})$, we define and study parabolic vector bundles on the quotient $\Sigma$ of the (extended) hyperbolic plane by $\Gamma$. If $\Gamma$ contains an…
We explore an elementary construction that produces finitely presented groups with diverse homological finiteness properties -- the {\em binary subgroups}, $B(\Sigma,\mu)<G_1\times\dots\times G_m$. These full subdirect products require…
We exhibit examples of finitely presented subgroups $P$ of direct products of hyperbolic groups for which there is no algorithm that detects whether a finitely presented group has a quotient isomorphic to $P$. For any torsion-free, linear,…