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Hard Metrics from Cayley Graphs of Abelian Groups
[chapter]

*
STACS 2007
*

In this paper we present a general method

doi:10.1007/978-3-540-70918-3_14
dblp:conf/stacs/NewmanR07
fatcat:k3goonscuzb5fduv6soodfjety
*of*constructing*hard**metrics*. Our results extend to embeddings into negative type*metric*spaces and into 1 . ...*Hard**metrics*are the class*of*extremal*metrics*with respect to embedding into Euclidean spaces: they incur Ω(log n) multiplicative distortion, which is as large as it can possibly get for any*metric*space ... It differs*from*that*of*[10, 3] , as the degree*of*such*Cayley**graphs*is necessarily non-constant. ...##
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Virtually Abelian quantum walks

2016
*
Journal of Physics A: Mathematical and Theoretical
*

We introduce quantum walks on

doi:10.1088/1751-8121/50/3/035301
fatcat:3totlpuqmvd7thergskvtnpt2m
*Cayley**graphs**of*non-*Abelian**groups*. ... We apply the technique in the case*of*two quantum walks on virtually*Abelian**groups*with planar*Cayley**graphs*, finding the exact solution. ... Quantum walks on*Cayley**graphs**of**Abelian**groups*The simplest case*of*quantum walks on*Cayley**graphs*is when the*group*is free*Abelian*, i.e. ...##
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Polynomial growth harmonic functions on finitely generated abelian groups

2013
*
Annals of Global Analysis and Geometry
*

While the

doi:10.1007/s10455-013-9374-0
fatcat:hxmueqd4hjfhhh5nt7h3ufdahm
*Cayley**graph*not only depends on the*abelian**group*, but also on the choice*of*a generating set, we find that this dimension depends only on the*group*itself. ... In the present paper, we develop geometric analytic techniques on*Cayley**graphs**of*finitely generated*abelian**groups*to study the polynomial growth harmonic functions. ... The*Cayley**graph**of*(G, S) is endowed with a natural*metric*, called the word*metric*(c.f. [6] ). ...##
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On Graphs, Groups and Geometry
[article]

2022
*
arXiv
*
pre-print

A

arXiv:2205.14097v1
fatcat:6fmpzkmuhzfrbcy4n466jw2iem
*group*is called natural if it emerges like this*from*a natural*metric*. A simple*graph*X is declared to be natural if (X,d) with geodesic*metric*d is natural. ... The semi-direct product*of*finite natural*groups*is natural too as they are represented by Zig-Zag products*of*suitable*Cayley**graphs*. It follows that wreath products preserve natural*groups*. ... The*metric*space emerges*from*a weighted zig-zag product*of*the corresponding weighted*Cayley**graphs*. ...##
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A characterization of cocompact hyperbolic and finite-volume hyperbolic groups in dimension three

1992
*
Transactions of the American Mathematical Society
*

We show that a cocompact hyperbolic

doi:10.1090/s0002-9947-1992-1036000-0
fatcat:2enbysgnsjeirir6tjh43m2t2i
*group*in dimension 3 is characterized by certain properties*of*its word*metric*which depend only on the*group*structure and not on any action on hyperbolic space. ... We prove a similar theorem for finite-volume hyperbolic*groups*in dimension 3. ... Instead*of*the*Cayley**graph*with word*metric*, we use an augmented*Cayley**graph*associated with a*group*G and a finite family*of*Euclidean subgroups*of*G. ...##
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A Characterization of Cocompact Hyperbolic and Finite-Volume Hyperbolic Groups in Dimension Three

1992
*
Transactions of the American Mathematical Society
*

We show that a cocompact hyperbolic

doi:10.2307/2154172
fatcat:jxvyrhmrj5ctnonodd7ebrrbwy
*group*in dimension 3 is characterized by certain properties*of*its word*metric*which depend only on the*group*structure and not on any action on hyperbolic space. ... We prove a similar theorem for finite-volume hyperbolic*groups*in dimension 3. ... Instead*of*the*Cayley**graph*with word*metric*, we use an augmented*Cayley**graph*associated with a*group*G and a finite family*of*Euclidean subgroups*of*G. ...##
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On isomorphisms of finite Cayley graphs—a survey

2002
*
Discrete Mathematics
*

The isomorphism problem for

doi:10.1016/s0012-365x(01)00438-1
fatcat:ru52e44jdncxxguuix64q45n4q
*Cayley**graphs*has been extensively investigated over the past 30 years. ... The methods used in this area range*from*deep*group*theory, including the ÿnite simple*group*classiÿcation, through to combinatorial techniques. ... overgroup PSL(2; 29), and is*hard*to read out*from*the elements*of*S. ...##
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A counterexample to the easy direction of the geometric Gersten conjecture

2019
*
Pacific Journal of Mathematics
*

For finitely generated

doi:10.2140/pjm.2019.298.27
fatcat:a4eigq7wa5b3zmjx7n7di4b65a
*groups*H and G, equipped with word*metrics*, a translation-like action*of*H on G is a free action such that each element*of*H acts by a map which has finite distance*from*the identity ... We show that the converse*of*this conjecture is false, and in particular the fundamental*group**of*a closed hyperbolic 3-manifold admits a translation-like action by the free*abelian**group**of*rank 2. ... action*of*H on G embeds the*Cayley**graph**of*H into some*Cayley**graph**of*G). ...##
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Research problems

1998
*
Discrete Mathematics
*

See [2] for the example

doi:10.1016/s0012-365x(98)00082-x
fatcat:kh5fhyiifjakfev7torvvupbhi
*of*the cycle*of*order 4 viewed as a*Cayley**graph*on the cyclic*group**of*order 4. ... Which*Cayley**graphs*on an*abelian**group*A can be isometrically embedded in the hypercube (up to scale) while preserving the MacWilliams duality for codes over the alphabet A? ...##
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The metric dimension of Cayley digraphs

2006
*
Discrete Mathematics
*

Moreover, the

doi:10.1016/j.disc.2005.09.015
fatcat:wjj4jwaaanbh5jct3mf4lsld6q
*metric*dimension*of*the*Cayley*digraph*of*the dihedral*group*D n*of*order 2n with a minimum set*of*generators is established. ... Sharp upper and lower bounds for the*metric*dimension*of*the*Cayley*digraphs Cay( : ), where is the*group*Z n 1 ⊕ Z n 2 ⊕ · · · ⊕ Z n m and is the canonical set*of*generators, are established. ... It was noted in [6] that the problem*of*finding the*metric*dimension*of*a*graph*is NP-*hard*. Khuller et al. [8] gave a construction that shows that the*metric*dimension*of*a*graph*is NP-*hard*. ...##
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The strong metric dimension of graphs and digraphs

2007
*
Discrete Applied Mathematics
*

Moreover, it is shown that computing this invariant is NP-

doi:10.1016/j.dam.2006.06.009
fatcat:drqglo6lyrbpjpbpgatsb6jmlq
*hard*. Related invariants for directed*graphs*are defined and studied. ... It is shown that the problem*of*finding the strong dimension*of*a connected*graph*can be transformed to the problem*of*finding the vertex covering number*of*a*graph*. ... For the remainder*of*this section we consider these invariants for*Cayley*(di)*graphs**of**abelian**groups*. It is known that every*abelian**group*is isomorphic to a direct product*of*cyclic*groups*. ...##
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Generalized Learning Problems and Applications to Non-commutative Cryptography
[chapter]

2011
*
Lecture Notes in Computer Science
*

It allows, for example, instantiations based on non-

doi:10.1007/978-3-642-24316-5_23
fatcat:wkjte6unwjfhrdrxcqy6u4bqwy
*abelian**groups*, resulting in a new avenue for the application*of*combinatorial*group*theory to the development*of*cryptographic primitives. ... We propose a generalization*of*the learning parity with noise (LPN) and learning with errors (LWE) problems to an abstract class*of**group*-theoretic learning problems that we term learning homomorphisms ... The maximum*Cayley*distance between any two elements in the*graph*is the diameter*of*the*Cayley**graph*. ...##
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Algebraic and computer-based methods in the undirected degree/diameter problem - A brief survey

English

2014
*
Electronic Journal of Graph Theory and Applications
*

English

Problem 1 (Degree/Diameter problem for undirected

doi:10.5614/ejgta.2014.2.2.9
fatcat:e57fyrwwojao3aayms3r2yz73y
*graphs*). Given positive integers ∆ and D, find the largest possible number*of*vertices N ∆,D*of*a*graph**of*maximum degree ∆ and diameter D. ... This paper discusses the most popular algebraic techniques and computational methods that have been used to construct large undirected*graphs*with given degree and diameter. ... The case*of**abelian**Cayley**graphs*is treated in depth in [26] . However, in order to obtain a large*Cayley**graph*, the*group*should be as far as possible*from**abelian*[32] . ...##
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The complexity of the weight problem for permutation and matrix groups

2010
*
Discrete Mathematics
*

Given a

doi:10.1016/j.disc.2009.03.005
fatcat:agtpcpzb5rbi3pgcw5ctudpfda
*metric*d on a permutation*group*G, the corresponding weight problem is to decide whether there exists an element π ∈ G such that d(π , e) = k, for some given value k. ... Here we show that this problem is NP-complete for many well-known*metrics*. ... Riis for his encouragement and kind support during the preparation*of*the paper. ...##
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On the scaling limit of finite vertex transitive graphs with large diameter
[article]

2014
*
arXiv
*
pre-print

Let (X_n) be an unbounded sequence

arXiv:1203.5624v4
fatcat:wixahh235rb4tp7ns4oyvhrchy
*of*finite, connected, vertex transitive*graphs*such that |X_n | = o(diam(X_n)^q) for some q>0. ... We show that up to taking a subsequence, and after rescaling by the diameter, the sequence (X_n) converges in the Gromov Hausdorff distance to a torus*of*dimension 1 sufficiently small, we prove, this ... We are grateful to László Pyber for noticing a mistake in a previous version*of*Lemma 2.5.4. He also mentioned to us some work in progress where some form*of*Lemma 2.5.4 is proved. ...
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