Murray Elder

On the Stanley--Wilf limit of 4231-avoiding permutations and a conjecture of Arratia

We show that the Stanley-Wilf limit for the class of 4231-avoiding permutations is at least by 9.47. This bound shows that this class has the largest such limit among all classes of permutations avoiding a single permutation of length 4 and refutes the conjecture that the Stanley-Wilf limit of a class of permutations avoiding a single permutation of length k cannot exceed (k-1)^2. The result is established by constructing a sequence of finite automata that accept subclasses of the class of 4231-avoiding permutations and analysing their transition matrices.

On groups that have normal forms computable in logspace

Abstract We consider the class of finitely generated groups which have a normal form computable in logspace . We prove that the class of such groups is closed under passing to finite index subgroups, direct products, wreath products, and certain free products and infinite extensions, and includes the solvable Baumslag–Solitar groups, as well as non-residually finite (and hence non-linear) examples. We define a group to be logspace embeddable if it embeds in a group with normal forms computable in logspace. We prove that finitely generated nilpotent groups are logspace embeddable. It follows that all groups of polynomial growth are logspace embeddable.

ON GROUPS WHOSE GEODESIC GROWTH IS POLYNOMIAL

This paper records some observations concerning geodesic growth functions. If a nilpotent group is not virtually cyclic then it has exponential geodesic growth with respect to all finite generating sets. On the other hand, if a finitely generated group G has an element whose normal closure is abelian and of finite index, then G has a finite generating set with respect to which the geodesic growth is polynomial (this includes all virtually cyclic groups).

A linear-time algorithm to compute geodesics in solvable Baumslag–Solitar groups

We present an algorithm to convert a word of length n in the standard generators of the solvable Baumslag–Solitar group BS(1, p) into a geodesic word, which runs in linear time and O(nlog n) space on a random access machine.

ON GROUPS AND COUNTER AUTOMATA

We study finitely generated groups whose word problems are accepted by counter automata. We show that a group has word problem accepted by a blind n-counter automaton in the sense of Greibach if and only if it is virtually free abelian of rank n; this result, which answers a question of Gilman, is in a very precise sense an abelian analogue of the Muller–Schupp theorem. More generally, if G is a virtually abelian group then every group with word problem recognized by a G-automaton is virtually abelian with growth class bounded above by the growth class of G. We consider also other types of counter automata.

Random subgroups of Thompson’s group F

We consider random subgroups of Thompsons group F with respect to two natural strati�cations of the set of all k generator subgroups. We �nd that the isomorphism classes of subgroups which occur with positive density are not the same for the two strati�cations. We give the �rst known exam- ples of persistent subgroups, whose isomorphism classes occur with positive density within the set of k-generator subgroups, for all su�ciently large k. Additionally, Thompsons group provides the �rst example of a group with- out a generic isomorphism class of subgroup. Elements of F are represented uniquely by reduced pairs of �nite rooted binary trees. We compute the asymptotic growth rate and a generating function for the number of reduced pairs of trees, which we show is D-�nite and not algebraic. We then use the asymptotic growth to prove our density results.

Minimal almost convexity

Abstract In this article we show that the Baumslag–Solitar group BS(1, 2) is minimally almost convex, or MAC. We also show that BS(1, 2) does not satisfy Poénaru’s almost convexity condition P(2), and hence the condition P(2) is strictly stronger than MAC. Finally, we show that the groups BS(1, q) for q ≥ 7 and Stallings’ non-FP3 group do not satisfy MAC. As a consequence, the condition MAC is not a commensurability invariant.

Curvature Testing in 3-Dimensional Metric Polyhedral Complexes

In a previous article, the authors described an algorithm to determine whether a finite metric polyhedral complex satisfied various local curvature conditions such as being locally CAT(0). The proof made use of Tarskis theorem about the decidability of first order sentences over the reals in an essential way, and thus it was not immediately applicable to a specific finite complex. In this article, we describe an algorithm restricted to 3-dimensional complexes which uses only elementary 3-dimensional geometry. After describing the procedure, we include several examples involving Euclidean tetrahedra which were run using an implementation of the algorithm in GAP.

Solution Sets for Equations over Free Groups are EDT0L Languages

We show that, given an equation over a finitely generated free group, the set of all solutions in reduced words forms an effectively constructible EDT0L language. In particular, the set of all solutions in reduced words is an indexed language in the sense of Aho. The language characterization we give, as well as further questions about the existence or finiteness of solutions, follow from our explicit construction of a finite directed graph which encodes all the solutions. Our result incorporates the recently invented recompression technique of Jez, and a new way to integrate solutions of linear Diophantine equations into the process. As a byproduct of our techniques, we improve the complexity from quadratic nondeterministic space in previous works to

CAT(0) IS AN ALGORITHMIC PROPERTY

\mathsf{NSPACE}(n\log n)