A Lower Estimate for Central Probabilities on Polycyclic Groups

1992 ◽  
Vol 44 (5) ◽  
pp. 897-910 ◽  
Author(s):  
G. Alexopoulos
Keyword(s):  
Probability Measure ◽  
Lie Group ◽  
Exponential Growth ◽  
Lower Estimate ◽  
Large Time ◽  
Polycyclic Group ◽  
Central Value ◽  
Convolution Power ◽  
Polycyclic Groups

AbstractWe give a lower estimate for the central value μ*n(e) of the nth convolution power μ*···*μ of a symmetric probability measure μ on a polycyclic group G of exponential growth whose support is finite and generates G. We also give a similar large time diagonal estimate for the fundamendal solution of the equation (∂/∂t + L)u = 0, where L is a left invariant sub-Laplacian on a unimodular amenable Lie group G of exponential growth.

scholarly journals The status of polycyclic group-based cryptography: A survey and open problems

2016 ◽  
Vol 8 (2) ◽  
Author(s):  
Jonathan Gryak ◽  
Delaram Kahrobaei
Keyword(s):  
Growth Rate ◽  
Exponential Growth ◽  
Polycyclic Group ◽  
Decision Problems ◽  
Exponential Growth Rate ◽  
Open Problems ◽  
Cyclic Groups ◽  
The Status ◽  
Polycyclic Groups ◽  
Finitely Presented

AbstractPolycyclic groups are natural generalizations of cyclic groups but with more complicated algorithmic properties. They are finitely presented and the word, conjugacy, and isomorphism decision problems are all solvable in these groups. Moreover, the non-virtually nilpotent ones exhibit an exponential growth rate. These properties make them suitable for use in group-based cryptography, which was proposed in 2004 by Eick and Kahrobaei [

scholarly journals Hyperconvex representations and exponential growth

2013 ◽  
Vol 34 (3) ◽  
pp. 986-1010 ◽  
Author(s):  
A. SAMBARINO
Keyword(s):  
Symmetric Space ◽  
Fundamental Group ◽  
Lie Group ◽  
Exponential Growth ◽  
Transversality Condition ◽  
Indicator Function ◽  
Equivariant Map ◽  
Negatively Curved ◽  
Funct Anal ◽  
Growth Indicator

AbstractLet $G$ be a real algebraic semi-simple Lie group and $\Gamma $ be the fundamental group of a closed negatively curved manifold. In this article we study the limit cone, introduced by Benoist [Propriétés asymptotiques des groupes linéaires. Geom. Funct. Anal. 7(1) (1997), 1–47], and the growth indicator function, introduced by Quint [Divergence exponentielle des sous-groupes discrets en rang supérieur. Comment. Math. Helv. 77 (2002), 503–608], for a class of representations $\rho : \Gamma \rightarrow G$ admitting an equivariant map from $\partial \Gamma $ to the Furstenberg boundary of the symmetric space of $G, $ together with a transversality condition. We then study how these objects vary with the representation.

On some polycyclic groups with small Hirsch length

2017 ◽  
Vol 16 (12) ◽  
pp. 1750237
Author(s):  
Heguo Liu ◽  
Fang Zhou ◽  
Tao Xu
Keyword(s):  
Abelian Subgroup ◽  
Polycyclic Group ◽  
Normal Abelian Subgroup ◽  
Finite Quotient ◽  
Polycyclic Groups

A polycyclic group [Formula: see text] is called an [Formula: see text]-group if every normal abelian subgroup of any finite quotient of [Formula: see text] is generated by [Formula: see text], or fewer, elements and [Formula: see text] is the least integer with this property. In this paper, the structure of [Formula: see text]-groups and [Formula: see text]-groups is determined.

POISSON SPACES FOR PROPER SEMIGROUPS OF SEMI-SIMPLE LIE GROUPS

2007 ◽  
Vol 07 (03) ◽  
pp. 273-297 ◽  
Author(s):  
JORGE N. LÓPEZ ◽  
PAULO R. C. RUFFINO ◽  
LUIZ A. B. SAN MARTIN
Keyword(s):  
Probability Measure ◽  
Lie Groups ◽  
Lie Group ◽  
Harmonic Functions ◽  
Classical Result ◽  
Continuous Functions ◽  
Empty Interior ◽  
Connected Group ◽  
Finite Center ◽  
Poisson Space

Let ν be a probability measure on a semi-simple Lie group G with finite center. Under the hypothesis that the semigroup S generated by ν has non-empty interior, we identify the Poisson space Π = G/MνAN, where bounded (l.u.c.) ν-harmonic functions in G have a one-to-one correspondence with measurable (continuous) functions in Π. This paper extends a classical result (see Furstenberg [7], Azencott [1] and others), where the semigroup generated by ν was assumed to be the whole (connected) group. We present two detailed examples.

On polycyclic groups with isomorphic finite quotients

1978 ◽  
Vol 84 (2) ◽  
pp. 235-246 ◽  
Author(s):  
Fritz Grunewald ◽  
Daniel Segal
Keyword(s):  
Finite Groups ◽  
Polycyclic Group ◽  
Isomorphism Classes ◽  
Finite Quotients ◽  
Polycyclic Groups ◽  
Special Case

Following P. F. Pickel (5) we write (G) for the set of isomorphism classes of finite quotients of a group G. One of the outstanding problems in the theory of polycyclic groups is to determine whether there can be infinitely many non-isomorphic polycyclic groups G with a given (G). We solve a special case of this problem with our first main result:Theorem 1. Let G be an abelian-by-cyclic polycyclic group. Then the polycyclic-by-finite groups H withlie in only finitely many isomorphism classes.

Convolution in perfect Lie groups

2016 ◽  
Vol 161 (1) ◽  
pp. 31-45
Author(s):  
YVES BENOIST ◽  
NICOLAS DE SAXCÉ
Keyword(s):  
Hausdorff Dimension ◽  
Lie Groups ◽  
Haar Measure ◽  
Lie Group ◽  
Borel Measure ◽  
Absolutely Continuous ◽  
Borel Set ◽  
Open Set ◽  
Convolution Power ◽  
Smooth Density

AbstractLetGbe a connected perfect real Lie group. We show that there exists α < dimGandp∈$\mathbb{N}$* such that if μ is a compactly supported α-Frostman Borel measure onG, then thepth convolution power μ*pis absolutely continuous with respect to the Haar measure onG, with arbitrarily smooth density. As an application, we obtain that ifA⊂Gis a Borel set with Hausdorff dimension at least α, then thep-fold product setApcontains a non-empty open set.

BREADTH IN POLYCYCLIC GROUPS

2007 ◽  
Vol 17 (05n06) ◽  
pp. 1073-1083 ◽  
Author(s):  
AVINOAM MANN ◽  
DAN SEGAL
Keyword(s):  
Finite Group ◽  
Group Theory ◽  
Nilpotent Group ◽  
Polycyclic Group ◽  
Finitely Generated ◽  
Finite Group Theory ◽  
Polycyclic Groups

The breadth of a polycyclic group is the maximum of h(G) - h(CG(x)) for x ∈ G, where h(G) is the Hirsch length. We prove a number of results that bound the class of a finitely generated nilpotent group, and the Hirsch length of the derived group in a polycyclic group, in terms of the breadth. These results are analogues of well-known results in finite group theory.

On some polycyclic groups with small Hirsch length II

2019 ◽  
Vol 18 (09) ◽  
pp. 1950169
Author(s):  
Heguo Liu ◽  
Fang Zhou ◽  
Tao Xu
Keyword(s):  
Abelian Subgroup ◽  
Polycyclic Group ◽  
Normal Abelian Subgroup ◽  
Number Of Generators ◽  
Finite Quotient ◽  
Polycyclic Groups

A polycyclic group [Formula: see text] is called a [Formula: see text]-group ([Formula: see text]-group) if every normal abelian subgroup (abelian subgroup) of any finite quotient of [Formula: see text] is generated by [Formula: see text], or fewer, elements and [Formula: see text] is the least integer with this property. In this paper, we describe the structures of [Formula: see text]-groups and [Formula: see text]-groups, and bound the number of generators of [Formula: see text]-groups and the derived lengths of [Formula: see text]-groups, which is a continuation of [H. G. Liu, F. Zhou and T. Xu, On some polycyclic groups with small Hirsch length, J. Algebra Appl. 16(11) (2017) 17502371–175023710].

The Polycyclic Length of Linear and Finite Polycyclic Groups

1974 ◽  
Vol 26 (4) ◽  
pp. 1002-1009 ◽  
Author(s):  
R. K. Fisher
Keyword(s):  
Polycyclic Group ◽  
Finite Series ◽  
Polycyclic Groups

In what follows, a polycyclic series, for the group G, is any finite seriesG = G0 ≧ G1 ≧ . . . ≧ Gl = 1of subgroups of G, such that Gi+1 ⊲ Gi and Gi/Gi+1 is cyclic, for all i = 0, . . ., l — 1. A group that has a polycyclic series is called a polycyclic group, and if G is a polycyclic group, then the polycyclic length of G, which we denote by ρ(G), is the number of non-trivial factors of a polycyclic series for G of shortest length.

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