TY - JOUR
T1 - Semi-direct product in groups and zig-zag product in graphs
T2 - Connections and applications
AU - Alon, Noga
AU - Lubotzky, Alexander
AU - Wigderson, Avi
N1 - Copyright:
Copyright 2017 Elsevier B.V., All rights reserved.
PY - 2001
Y1 - 2001
N2 - We consider the standard semi-direct product A × B of finite groups A, B. We show that with certain choices of generators for these three groups, the Cayley graph of A × B is (essentially) the zigzag product of the Cayley graphs of A and B. Thus, using the results of [RVW00], the new Cayley graph is an expander if and only if its two components are. We develop some general ways of using this construction to obtain large constant-degree expanding Cayley graphs from small ones. In [LW93], Lubotzky and Weiss asked whether expansion is a group property; namely, is being expander for (a Cayley graph of) a group G depend solely on G and not on the choice of generators. We use the above construction to answer the question in negative, by showing an infinite family of groups Ai × Bi which are expanders with one choice of (constant-size) set of generators and are not with another such choice. It is interesting to note that this problem is still open, though, for "natural" families of groups, like the symmetric groups Sn or the simple groups PSL(2, p).
AB - We consider the standard semi-direct product A × B of finite groups A, B. We show that with certain choices of generators for these three groups, the Cayley graph of A × B is (essentially) the zigzag product of the Cayley graphs of A and B. Thus, using the results of [RVW00], the new Cayley graph is an expander if and only if its two components are. We develop some general ways of using this construction to obtain large constant-degree expanding Cayley graphs from small ones. In [LW93], Lubotzky and Weiss asked whether expansion is a group property; namely, is being expander for (a Cayley graph of) a group G depend solely on G and not on the choice of generators. We use the above construction to answer the question in negative, by showing an infinite family of groups Ai × Bi which are expanders with one choice of (constant-size) set of generators and are not with another such choice. It is interesting to note that this problem is still open, though, for "natural" families of groups, like the symmetric groups Sn or the simple groups PSL(2, p).
UR - http://www.scopus.com/inward/record.url?scp=0035181247&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=0035181247&partnerID=8YFLogxK
U2 - 10.1109/SFCS.2001.959939
DO - 10.1109/SFCS.2001.959939
M3 - Article
AN - SCOPUS:0035181247
SN - 0272-5428
SP - 630
EP - 637
JO - Annual Symposium on Foundations of Computer Science - Proceedings
JF - Annual Symposium on Foundations of Computer Science - Proceedings
ER -