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Hard Metrics from Cayley Graphs of Abelian Groups [chapter]

Ilan Newman, Yuri Rabinovich
STACS 2007  
In this paper we present a general method 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.  ... 
doi:10.1007/978-3-540-70918-3_14 dblp:conf/stacs/NewmanR07 fatcat:k3goonscuzb5fduv6soodfjety

Virtually Abelian quantum walks

Giacomo Mauro D'Ariano, Marco Erba, Paolo Perinotti, Alessandro Tosini
2016 Journal of Physics A: Mathematical and Theoretical  
We introduce quantum walks on 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.  ... 
doi:10.1088/1751-8121/50/3/035301 fatcat:3totlpuqmvd7thergskvtnpt2m

Polynomial growth harmonic functions on finitely generated abelian groups

Bobo Hua, Jürgen Jost, Xianqing Li-Jost
2013 Annals of Global Analysis and Geometry  
While the 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] ).  ... 
doi:10.1007/s10455-013-9374-0 fatcat:hxmueqd4hjfhhh5nt7h3ufdahm

On Graphs, Groups and Geometry [article]

Oliver Knill
2022 arXiv   pre-print
A 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.  ... 
arXiv:2205.14097v1 fatcat:6fmpzkmuhzfrbcy4n466jw2iem

A characterization of cocompact hyperbolic and finite-volume hyperbolic groups in dimension three

J. W. Cannon, Daryl Cooper
1992 Transactions of the American Mathematical Society  
We show that a cocompact hyperbolic 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.  ... 
doi:10.1090/s0002-9947-1992-1036000-0 fatcat:2enbysgnsjeirir6tjh43m2t2i

A Characterization of Cocompact Hyperbolic and Finite-Volume Hyperbolic Groups in Dimension Three

J. W. Cannon, Daryl Cooper
1992 Transactions of the American Mathematical Society  
We show that a cocompact hyperbolic 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.  ... 
doi:10.2307/2154172 fatcat:jxvyrhmrj5ctnonodd7ebrrbwy

On isomorphisms of finite Cayley graphs—a survey

Cai Heng Li
2002 Discrete Mathematics  
The isomorphism problem for 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.  ... 
doi:10.1016/s0012-365x(01)00438-1 fatcat:ru52e44jdncxxguuix64q45n4q

A counterexample to the easy direction of the geometric Gersten conjecture

David Cohen
2019 Pacific Journal of Mathematics  
For finitely generated 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).  ... 
doi:10.2140/pjm.2019.298.27 fatcat:a4eigq7wa5b3zmjx7n7di4b65a

Research problems

1998 Discrete Mathematics  
See [2] for the example 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?  ... 
doi:10.1016/s0012-365x(98)00082-x fatcat:kh5fhyiifjakfev7torvvupbhi

The metric dimension of Cayley digraphs

Melodie Fehr, Shonda Gosselin, Ortrud R. Oellermann
2006 Discrete Mathematics  
Moreover, the 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.  ... 
doi:10.1016/j.disc.2005.09.015 fatcat:wjj4jwaaanbh5jct3mf4lsld6q

The strong metric dimension of graphs and digraphs

Ortrud R. Oellermann, Joel Peters-Fransen
2007 Discrete Applied Mathematics  
Moreover, it is shown that computing this invariant is NP-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.  ... 
doi:10.1016/j.dam.2006.06.009 fatcat:drqglo6lyrbpjpbpgatsb6jmlq

Generalized Learning Problems and Applications to Non-commutative Cryptography [chapter]

Gilbert Baumslag, Nelly Fazio, Antonio R. Nicolosi, Vladimir Shpilrain, William E. Skeith
2011 Lecture Notes in Computer Science  
It allows, for example, instantiations based on non-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.  ... 
doi:10.1007/978-3-642-24316-5_23 fatcat:wkjte6unwjfhrdrxcqy6u4bqwy

Algebraic and computer-based methods in the undirected degree/diameter problem - A brief survey
English

Hebert Perez-Roses
2014 Electronic Journal of Graph Theory and Applications  
Problem 1 (Degree/Diameter problem for undirected 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] .  ... 
doi:10.5614/ejgta.2014.2.2.9 fatcat:e57fyrwwojao3aayms3r2yz73y

The complexity of the weight problem for permutation and matrix groups

Peter J. Cameron, Taoyang Wu
2010 Discrete Mathematics  
Given a 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.  ... 
doi:10.1016/j.disc.2009.03.005 fatcat:agtpcpzb5rbi3pgcw5ctudpfda

On the scaling limit of finite vertex transitive graphs with large diameter [article]

Itai Benjamini, Hilary Finucane, Romain Tessera
2014 arXiv   pre-print
Let (X_n) be an unbounded sequence 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.  ... 
arXiv:1203.5624v4 fatcat:wixahh235rb4tp7ns4oyvhrchy
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