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Search Results: 1 - 10 of 2721 matches for " Mikhail Belolipetsky "
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On fields of definition of arithmetic Kleinian reflection groups
Mikhail Belolipetsky
Mathematics , 2007,
Abstract: We show that degrees of the real fields of definition of arithmetic Kleinian reflection groups are bounded by 35.
On 2-systoles of hyperbolic 3-manifolds
Mikhail Belolipetsky
Mathematics , 2012,
Abstract: We investigate the geometry of $\pi_1$-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 $\pi_1$-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 $\pi_1(M_i)$ 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.
Hyperbolic orbifolds of small volume
Mikhail Belolipetsky
Mathematics , 2014,
Abstract: 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.
Geodesics, volumes and Lehmer's conjecture
Mikhail Belolipetsky
Mathematics , 2011,
Abstract: 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.
Arithmetic hyperbolic reflection groups
Mikhail Belolipetsky
Mathematics , 2015,
Abstract: This is a survey article about arithmetic hyperbolic reflection groups with an emphasis on the results that were obtained in the last ten years and on the open problems.
Finiteness theorems for congruence reflection groups
Mikhail Belolipetsky
Mathematics , 2010,
Abstract: 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.
Reflective and quasi-reflective Bianchi groups
Mikhail Belolipetsky,John Mcleod
Mathematics , 2012,
Abstract: 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.
On fields of definition of arithmetic Kleinian reflection groups II
Mikhail Belolipetsky,Benjamin Linowitz
Mathematics , 2012,
Abstract: 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.
Manifolds counting and class field towers
Mikhail Belolipetsky,Alex Lubotzky
Mathematics , 2009,
Abstract: 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 $x^{(\gamma(H)+o(1))\log x/\log\log x}$ where $\gamma(H)$ 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.
On volumes of arithmetic quotients of PO(n,1), n odd
Mikhail Belolipetsky,Vincent Emery
Mathematics , 2010, DOI: 10.1112/plms/pds009
Abstract: 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.
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