12 research outputs found
The Power Word Problem
In this work we introduce a new succinct variant of the word problem in a finitely generated group G, which we call the power word problem: the input word may contain powers p^x, where p is a finite word over generators of G and x is a binary encoded integer. The power word problem is a restriction of the compressed word problem, where the input word is represented by a straight-line program (i.e., an algebraic circuit over G). The main result of the paper states that the power word problem for a finitely generated free group F is AC^0-Turing-reducible to the word problem for F. Moreover, the following hardness result is shown: For a wreath product G Wr Z, where G is either free of rank at least two or finite non-solvable, the power word problem is complete for coNP. This contrasts with the situation where G is abelian: then the power word problem is shown to be in TC^0
The Complexity of Knapsack Problems in Wreath Products
We prove new complexity results for computational problems in certain wreath
products of groups and (as an application) for free solvable group. For a
finitely generated group we study the so-called power word problem (does a
given expression , where are
words over the group generators and are binary encoded
integers, evaluate to the group identity?) and knapsack problem (does a given
equation , where are words
over the group generators and are variables, has a solution in
the natural numbers). We prove that the power word problem for wreath products
of the form with nilpotent and iterated wreath products
of free abelian groups belongs to . As an application of the
latter, the power word problem for free solvable groups is in .
On the other hand we show that for wreath products , where
is a so called uniformly strongly efficiently non-solvable group (which
form a large subclass of non-solvable groups), the power word problem is
-hard. For the knapsack problem we show
-completeness for iterated wreath products of free abelian groups
and hence free solvable groups. Moreover, the knapsack problem for every wreath
product , where is uniformly efficiently non-solvable, is
-hard
Quadratic equations in metabelian Baumslag-Solitar groups
For a finitely generated group , the Diophantine problem over is the
algorithmic problem of deciding whether a given equation (perhaps restricted to a fixed subclass of equations) has a solution in
. We investigate the algorithmic complexity of the Diophantine problem for
the class of quadratic equations over the metabelian
Baumslag-Solitar groups . In particular, we prove that this problem is
NP-complete whenever , and determine the algorithmic complexity for
various subclasses (orientable, nonorientable etc.) of .Comment: 17 page
A Characterization of Wreath Products Where Knapsack Is Decidable
The knapsack problem for groups was introduced by Miasnikov, Nikolaev, and
Ushakov. It is defined for each finitely generated group and takes as input
group elements and asks whether there are
with . We study the knapsack
problem for wreath products of groups and . Our main result is
a characterization of those wreath products for which the knapsack
problem is decidable. The characterization is in terms of decidability
properties of the indiviual factors and . To this end, we introduce two
decision problems, the intersection knapsack problem and its restriction, the
positive intersection knapsack problem. Moreover, we apply our main result to
, the discrete Heisenberg group, and to Baumslag-Solitar
groups for . First, we show that the knapsack
problem is undecidable for for any . This
implies that for and for infinite and virtually nilpotent groups ,
the knapsack problem for is decidable if and only if is virtually
abelian and solvability of systems of exponent equations is decidable for .
Second, we show that the knapsack problem is decidable for
if and only if solvability of systems of exponent
equations is decidable for
Subgroup Membership in GL(2,Z)
It is shown that the subgroup membership problem for a virtually free group can be decided in polynomial time where all group elements are represented by so-called power words, i.e., words of the form p_1^{z_1} p_2^{z_2} ? p_k^{z_k}. Here the p_i are explicit words over the generating set of the group and all z_i are binary encoded integers. As a corollary, it follows that the subgroup membership problem for the matrix group GL(2,?) can be decided in polynomial time when all matrix entries are given in binary notation
Exponent equations in HNN-extensions
We consider exponent equations in finitely generated groups. These are
equations, where the variables appear as exponents of group elements and take
values from the natural numbers. Solvability of such (systems of) equations has
been intensively studied for various classes of groups in recent years. In many
cases, it turns out that the set of all solutions on an exponent equation is a
semilinear set that can be constructed effectively. Such groups are called
knapsack semilinear. Examples of knapsack semilinear groups are hyperbolic
groups, virtually special groups, co-context-free groups and free solvable
groups. Moreover, knapsack semilinearity is preserved by many group theoretic
constructions, e.g., finite extensions, graph products, wreath products,
amalgamated free products with finite amalgamated subgroups, and HNN-extensions
with finite associated subgroups. On the other hand, arbitrary HNN-extensions
do not preserve knapsack semilinearity. In this paper, we consider the knapsack
semilinearity of HNN-extensions, where the stable letter acts trivially by
conjugation on the associated subgroup of the base group . We show that
under some additional technical conditions, knapsack semilinearity transfers
from base group to the HNN-extension . These additional technical
conditions are satisfied in many cases, e.g., when is a centralizer in
or is a quasiconvex subgroup of the hyperbolic group .Comment: A short version appeared in Proceedings of ISSAC 202