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Mikhail Belolipetsky

Publications and source records attributed to Mikhail Belolipetsky.

34 records · Page 2Linked to original sources
arXiv2016

Singular sets and parameters of generalized triangle orbifolds

We study groups generated by three half-turns in the Lobachevsky $3$-space and their quotient orbifolds. These generalized triangle groups are closely related to the arbitrary 2-generator Kleinian groups. Our main result is a classification of the singular sets of the generalized triangle orbifolds. We also present a method to obtain the parameters defining a generalized triangle group from the structure of the singular set of its quotient orbifold and illustrate it by examples.

math.MG↗
arXiv2016

Arithmetic hyperbolic reflection groups

A hyperbolic reflection group is a discrete group generated by reflections in the faces of an $n$-dimensional hyperbolic polyhedron. This survey article is dedicated to the study of arithmetic hyperbolic reflection groups with an emphasis on the results that were obtained in the last ten years and on the open problems.

math.GT↗
arXiv2014

Kazhdan-Lusztig cells in planar hyperbolic Coxeter groups and automata

Let C be a one- or two-sided Kazhdan--Lusztig cell in a Coxeter group (W,S), and let Reduced(C) denote the set of reduced expressions of all w in C, regarded as a language over the alphabet S. Casselman has conjectured that Reduced(C) is regular. In this paper we give a conjectural description of the cells when W is the group corresponding to a hyperbolic polygon, and show that our conjectures imply Casselman's.

math.RT↗
arXiv2014

Hyperbolic orbifolds of small volume

Volume is a natural measure of complexity of a Riemannian manifold. In this survey, we discuss the results and conjectures concerning n-dimensional hyperbolic manifolds and orbifolds of small volume.

math.MG↗
arXiv2013

Hyperbolic manifolds of small volume

We conjecture that for every dimension n not equal 3 there exists a noncompact hyperbolic n-manifold whose volume is smaller than the volume of any compact hyperbolic n-manifold. For dimensions n at most 4 and n=6 this conjecture follows from the known results. In this paper we show that the conjecture is true for arithmetic hyperbolic n-manifolds of dimension n at least 30.

math.MG↗
arXiv2013

Reflective and quasi-reflective Bianchi groups

A discrete subgroup of the group of isometries of the hyperbolic space is called reflective if up to a finite index it is generated by reflections in hyperplanes. The main result of this paper is a complete classification of the reflective (and quasi-reflective) subgroups among the Bianchi groups and their extensions.

math.GR↗
arXiv2013

On fields of definition of arithmetic Kleinian reflection groups II

Following the previous work of Nikulin and Agol, Belolipetsky, Storm, and Whyte it is known that there exist only finitely many (totally real) number fields that can serve as fields of definition of arithmetic hyperbolic reflection groups. We prove a new bound on the degree $n_k$ of these fields in dimension 3: $n_k$ does not exceed 9. Combined with previous results of Maclachlan and Nikulin, this leads to a new bound $n_k \le 25$ which is valid for all dimensions. We also obtain upper bounds for the discriminants of these fields and give some heuristic results which may be useful for the classification of arithmetic hyperbolic reflection groups.

math.GT↗
arXiv2012

On 2-systoles of hyperbolic 3-manifolds

We investigate the geometry of $π_1$-injective surfaces in closed hyperbolic 3-manifolds. First we prove that for any $e>0$, if the manifold $M$ has sufficiently large systole $\sys_1(M)$, the genus of any such surface in $M$ is bounded below by $\exp((1/2-e)\sys_1(M))$. Using this result we show, in particular, that for congruence covers $M_i\to M$ of a compact arithmetic hyperbolic 3-manifold we have: (a) the minimal genus of $π_1$-injective surfaces satisfies $\log \sysg(M_i) \gtrsim (1/3)\log\vol(M_i)$; (b) there exist such sequences with the ratio Heegard genus$(M_i)/\sysg(M_i) \gtrsim \vol(M_i)^{1/2}$; and (c) under some additional assumptions $π_1(M_i)$ is k-free with $\log k \gtrsim (1/3)\sys_1(M_i)$. The latter resolves a special case of a conjecture of M. Gromov.

math.GT↗
arXiv2011

On volumes of arithmetic quotients of PO(n,1), n odd

We determine the minimal volume of arithmetic hyperbolic orientable n-dimensional orbifolds (compact and non-compact) for every odd dimension n>3. Combined with the previously known results it solves the minimal volume problem for arithmetic hyperbolic n-orbifolds in all dimensions.

math.GR↗
arXiv2011

Manifolds counting and class field towers

In [BGLM] and [GLNP] it was conjectured that if $H$ is a simple Lie group of real rank at least 2, then the number of conjugacy classes of (arithmetic) lattices in $H$ of covolume at most $x$ is $x^{(γ(H)+o(1))\log x/\log\log x}$ where $γ(H)$ is an explicit constant computable from the (absolute) root system of $H$. In this paper we prove that this conjecture is false. In fact, we show that the growth is at rate $x^{c\log x}$. A crucial ingredient of the proof is the existence of towers of field extensions with bounded root discriminant which follows from the seminal work of Golod and Shafarevich on class field towers.

math.GR↗
arXiv2011

Geodesics, volumes and Lehmer's conjecture

In this report I discuss the relations between systoles and volumes of hyperbolic manifolds and a conjecture of Lehmer about the Mahler measure of non-cyclotomic polynomials.

math.GT↗
arXiv2011

Systoles of Hyperbolic Manifolds

We show that for every $n\geq 2$ and any $ε>0$ there exists a compact hyperbolic $n$-manifold with a closed geodesic of length less than $ε$. When $ε$ is sufficiently small these manifolds are non-arithmetic, and they are obtained by a generalised inbreeding construction which was first suggested by Agol for $n=4$. We also show that for $n\geq 3$ the volumes of these manifolds grow at least as $1/ε^{n-2}$ when $ε\to 0$.

math.GT↗
arXiv2011

Finiteness theorems for congruence reflection groups

This paper is a follow-up to our joint paper with I. Agol, P. Storm and K. Whyte "Finiteness of arithmetic hyperbolic reflection groups". The main purpose is to investigate the effective side of the method developed there and its possible application to the problem of classification of arithmetic hyperbolic reflection groups.

math.GT↗
arXiv2010

Counting arithmetic lattices and surfaces

We give estimates on the number $AL_H(x)$ of arithmetic lattices $Γ$ of covolume at most $x$ in a simple Lie group $H$. In particular, we obtain a first concrete estimate on the number of arithmetic 3-manifolds of volume at most $x$. Our main result is for the classical case $H=PSL(2,R)$ where we compute the limit of $\log AL_H(x) / x\log x$ when $x\to\infty$. The proofs use several different techniques: geometric (bounding the number of generators of $Γ$ as a function of its covolume), number theoretic (bounding the number of maximal such $Γ$) and sharp estimates on the character values of the symmetric groups (to bound the subgroup growth of $Γ$).

math.GR↗