Searcharxiv⌕ Search

arXiv subjects

Nikolay Nikolov

Publications and source records attributed to Nikolay Nikolov.

104 records · Page 6Linked to original sources
arXiv2011

On the growth of Betti numbers of locally symmetric spaces

We announce new results concerning the asymptotic behavior of the Betti numbers of higher rank locally symmetric spaces as their volumes tend to infinity. Our main theorem is a uniform version of the Lück Approximation Theorem \cite{luck}, which is much stronger than the linear upper bounds on Betti numbers given by Gromov in \cite{BGS}. The basic idea is to adapt the theory of local convergence, originally introduced for sequences of graphs of bounded degree by Benjamimi and Schramm, to sequences of Riemannian manifolds. Using rigidity theory we are able to show that when the volume tends to infinity, the manifolds locally converge to the universal cover in a sufficiently strong manner that allows us to derive the convergence of the normalized Betti numbers.

math.GR↗
arXiv2011

The rank gradient from a combinatorial viewpoint

This paper investigates the asymptotic behaviour of the minimal number of generators of finite index subgroups in residually finite groups. We analyze three natural classes of groups: amenable groups, groups possessing an infinite soluble normal subgroup and virtually free groups. As a tool for the amenable case we generalize Lackenby's trichotomy theorem on finitely presented groups.

math.GR↗
arXiv2009

Powers in finite groups

In this note we prove that if $G$ is a finitely generated profinite group then the verbal subgroup $G^{q}$ is open. Equivalently in a $d$-generator finite group every product of $q$th powers is a product of $f(d,q)$ $q$th powers.

math.GR↗
arXiv2008

Rank gradient, cost of groups and the rank versus Heegaard genus problem

We study the growth of the rank of subgroups of finite index in residually finite groups, by relating it to the notion of cost. As a by-product, we show that the `Rank vs. Heegaard genus' conjecture on hyperbolic 3-manifolds is incompatible with the `Fixed Price problem' in topological dynamics.

math.GR↗
arXiv2008

Generation of polycyclic groups

In this note we give an alternative proof of a theorem of Linnell and Warhurst that the number of generators d(G) of a polycyclic group G is at most d(\hat G), where d(\hat G) is the number of generators of the profinite completion of G. While not claiming anything new we believe that our argument is much simpler that the original one. Moreover our result gives some sufficient condition when d(G)=d(\hat G) which can be verified quite easily in the case when G is virtually abelian.

math.GR↗
arXiv2007

Product decompositions of quasirandom groups and a Jordan type theorem

We first note that a result of Gowers on product-free sets in groups has an unexpected consequence: If k is the minimal degree of a representation of the finite group G, then for every subset B of G with $|B| > |G| / k^{1/3}$ we have B^3 = G. We use this to obtain improved versions of recent deep theorems of Helfgott and of Shalev concerning product decompositions of finite simple groups, with much simpler proofs. On the other hand, we prove a version of Jordan's theorem which implies that if k>1, then G has a proper subgroup of index at most ck^2 for some absolute constant c, hence a product-free subset of size at least $|G| / c'k$. This answers a question of Gowers.

math.GR↗
arXiv2006

Unitary Positive-Energy Representations of Scalar Bilocal Quantum Fields

The superselection sectors of two classes of scalar bilocal quantum fields in D>=4 dimensions are explicitly determined by working out the constraints imposed by unitarity. The resulting classification in terms of the dual of the respective gauge groups U(N) and O(N) confirms the expectations based on general results obtained in the framework of local nets in algebraic quantum field theory, but the approach using standard Lie algebra methods rather than abstract duality theory is complementary. The result indicates that one does not lose interesting models if one postulates the absence of scalar fields of dimension D-2 in models with global conformal invariance. Another remarkable outcome is the observation that, with an appropriate choice of the Hamiltonian, a Lie algebra embedded into the associative algebra of observables completely fixes the representation theory.

math-ph↗
arXiv2006

On finitely generated profinite groups I: strong completeness and uniform bounds

We prove that in every finitely generated profinite group, every subgroup of finite index is open; this implies that the topology on such groups is determined by the algebraic structure. This is deduced from the main result about finite groups: let $w$ be a `locally finite' group word and $d\in\mathbb{N}$. Then there exists $f=f(w,d)$ such that in every $d$-generator finite group $G$, every element of the verbal subgroup $w(G)$ is equal to a product of $f$ $w$-values. An analogous theorem is proved for commutators; this implies that in every finitely generated profinite group, each term of the lower central series is closed. The proofs rely on some properties of the finite simple groups, to be established in Part II.

math.GR↗
arXiv2006

On finitely generated profinite groups II, products in quasisimple groups

We prove two results. (1) There is an absolute constant $D$ such that for any finite quasisimple group $S$, given 2D arbitrary automorphisms of $S$, every element of $S$ is equal to a product of $D$ `twisted commutators' defined by the given automorphisms. (2) Given a natural number $q$, there exist $C=C(q)$ and $M=M(q)$ such that: if $S$ is a finite quasisimple group with $| S/\mathrm{Z}(S)| >C$, $β_{j}$ $ (j=1,...,M)$ are any automorphisms of $S$, and $q_{j}$ $ (j=1,...,M)$ are any divisors of $q$, then there exist inner automorphisms $α_{j}$ of $S$ such that $S=\prod_{1}^{M}[S,(α_{j}β_{j})^{q_{j}}]$. These results, which rely on the Classification of finite simple groups, are needed to complete the proofs of the main theorems of Part I.

math.GR↗
arXiv2006

Cartesian products as profinite completions

We prove that if a Cartesian product of alternating groups is topologically finitely generated, then it is the profinite completion of a finitely generated residually finite group. The same holds for Cartesian producs of other simple groups under some natural restrictions.

math.GR↗
arXiv2005

Finite Simple Groups as Expanders

We prove that there exist $k\in N$ and $0<ε\in R$ such that every non-abelian finite simple group $G$, which is not a Suzuki group, has a set of $k$ generators for which the Cayley graph $\Cay(G; S)$ is an $ε$-expander.

math.GR↗
arXiv2005

Universal lattices and Property $τ$

We prove that the universal lattices -- the groups $G=\SL_d(R)$ where $R=\Z[x_1,...,x_k]$, have property $τ$ for $d\geq 3$. This provides the first example of linear groups with $τ$ which do not come from arithmetic groups. We also give a lower bound for the expanding constant with respect to the natural generating set of $G$. Our methods are based on bounded elementary generation of the finite congruence images of $G$, a generalization of a result by Dennis and Stein on $K_2$ of some finite commutative rings and a relative property \emph{T} of $(\SL_2(R) \ltimes R^2, R^2)$.

math.GR↗