9 research outputs found
Generic-case complexity, decision problems in group theory and random walks
We give a precise definition of ``generic-case complexity'' and show that for
a very large class of finitely generated groups the classical decision problems
of group theory - the word, conjugacy and membership problems - all have
linear-time generic-case complexity. We prove such theorems by using the theory
of random walks on regular graphs.Comment: Revised versio
Conjugacy in Baumslag's group, generic case complexity, and division in power circuits
The conjugacy problem belongs to algorithmic group theory. It is the
following question: given two words x, y over generators of a fixed group G,
decide whether x and y are conjugated, i.e., whether there exists some z such
that zxz^{-1} = y in G. The conjugacy problem is more difficult than the word
problem, in general. We investigate the complexity of the conjugacy problem for
two prominent groups: the Baumslag-Solitar group BS(1,2) and the
Baumslag(-Gersten) group G(1,2). The conjugacy problem in BS(1,2) is
TC^0-complete. To the best of our knowledge BS(1,2) is the first natural
infinite non-commutative group where such a precise and low complexity is
shown. The Baumslag group G(1,2) is an HNN-extension of BS(1,2). We show that
the conjugacy problem is decidable (which has been known before); but our
results go far beyond decidability. In particular, we are able to show that
conjugacy in G(1,2) can be solved in polynomial time in a strongly generic
setting. This means that essentially for all inputs conjugacy in G(1,2) can be
decided efficiently. In contrast, we show that under a plausible assumption the
average case complexity of the same problem is non-elementary. Moreover, we
provide a lower bound for the conjugacy problem in G(1,2) by reducing the
division problem in power circuits to the conjugacy problem in G(1,2). The
complexity of the division problem in power circuits is an open and interesting
problem in integer arithmetic.Comment: Section 5 added: We show that an HNN extension G = < H, b | bab^-1 =
{\phi}(a), a \in A > has a non-amenable Schreier graph with respect to the
base group H if and only if A \neq H \neq
Multicoloured Random Graphs: Constructions and Symmetry
This is a research monograph on constructions of and group actions on
countable homogeneous graphs, concentrating particularly on the simple random
graph and its edge-coloured variants. We study various aspects of the graphs,
but the emphasis is on understanding those groups that are supported by these
graphs together with links with other structures such as lattices, topologies
and filters, rings and algebras, metric spaces, sets and models, Moufang loops
and monoids. The large amount of background material included serves as an
introduction to the theories that are used to produce the new results. The
large number of references should help in making this a resource for anyone
interested in beginning research in this or allied fields.Comment: Index added in v2. This is the first of 3 documents; the other 2 will
appear in physic