38 research outputs found
A context-free and a 1-counter geodesic language for a Baumslag-Solitar group
We give a language of unique geodesic normal forms for the Baumslag-Solitar
group BS(1,2) that is context-free and 1-counter. We discuss the classes of
context-free, 1-counter and counter languages, and explain how they are
inter-related
C-graph automatic groups
We generalize the notion of a graph automatic group introduced by Kharlampovich, Khoussainov and Miasnikov by replacing the regular languages in their definition with more powerful language classes. For a fixed language class C, we call the resulting groups C-graph automatic. We prove that the class of C-graph automatic groups is closed under change of generating set, direct and free product for certain classes C. We show that for quasi-realtime counter-graph automatic groups where normal forms have length that is linear in the geodesic length, there is an algorithm to compute normal forms (and therefore solve the word problem) in polynomial time. The class of quasi-realtime counter-graph automatic groups includes all Baumslag-Solitar groups, and the free group of countably infinite rank. Context-sensitive-graph automatic groups are shown to be a very large class, which encompasses, for example, groups with unsolvable conjugacy problem, the Grigorchuk group, and Thompson's groups F, T and V. © 2014 Elsevier Inc
Metric properties of Baumslag-Solitar groups
© 2015 World Scientific Publishing Company. We compute estimates for the word metric of Baumslag-Solitar groups in terms of the Britton's lemma normal form. As a corollary, we find lower bounds for the growth rate for the groups BS(p, q) with 1 < p ≤ q
Limits of Baumslag-Solitar groups and dimension estimates in the space of marked groups
We prove that the limits of Baumslag-Solitar groups which we previously
studied are non-linear hopfian C*-simple groups with infinitely many twisted
conjugacy classes. We exhibit infinite presentations for these groups, classify
them up to group isomorphism, describe their automorphisms and discuss the word
and conjugacy problems. Finally, we prove that the set of these groups has
non-zero Hausforff dimension in the space of marked groups on two generators.Comment: 30 pages, no figures, englis
Languages, groups and equations
The survey provides an overview of the work done in the last 10 years to
characterise solutions to equations in groups in terms of formal languages. We
begin with the work of Ciobanu, Diekert and Elder, who showed that solutions to
systems of equations in free groups in terms of reduced words are expressible
as EDT0L languages. We provide a sketch of their algorithm, and describe how
the free group results extend to hyperbolic groups. The characterisation of
solutions as EDT0L languages is very robust, and many group constructions
preserve this, as shown by Levine.
The most recent progress in the area has been made for groups without
negative curvature, such as virtually abelian, the integral Heisenberg group,
or the soluble Baumslag-Solitar groups, where the approaches to describing the
solutions are different from the negative curvature groups. In virtually
abelian groups the solutions sets are in fact rational, and one can obtain them
as -regular sets. In the Heisenberg group producing the solutions to a
single equation reduces to understanding the solutions to quadratic Diophantine
equations and uses number theoretic techniques. In the Baumslag-Solitar groups
the methods are combinatorial, and focus on the interplay of normal forms to
solve particular classes of equations.
In conclusion, EDT0L languages give an effective and simple combinatorial
characterisation of sets of seemingly high complexity in many important classes
of groups.Comment: 26 page
Groups whose word problems are not semilinear
Suppose that G is a finitely generated group and W is the formal language of
words defining the identity in G. We prove that if G is a nilpotent group, the
fundamental group of a finite volume hyperbolic three-manifold, or a
right-angled Artin group whose graph lies in a certain infinite class, then W
is not a multiple context free language
Regular left-orders on groups
A regular left-order on finitely generated group is a total,
left-multiplication invariant order on whose corresponding positive cone is
the image of a regular language over the generating set of the group under the
evaluation map. We show that admitting regular left-orders is stable under
extensions and wreath products and give a classification of the groups all
whose left-orders are regular left-orders. In addition, we prove that solvable
Baumslag-Solitar groups admits a regular left-order if and only if
. Finally, Hermiller and Sunic showed that no free product admits a
regular left-order, however we show that if and are groups with regular
left-orders, then admits a regular left-order.Comment: 41 pages,9 figure
The word problem distinguishes counter languages
Counter automata are more powerful versions of finite-state automata where
addition and subtraction operations are permitted on a set of n integer
registers, called counters. We show that the word problem of is accepted
by a nondeterministic -counter automaton if and only if .Comment: 8 page