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Murray Elder

Publications and source records attributed to Murray Elder.

At least 55 records · Page 3Linked to original sources
arXiv2014

$\mathcal C$-graph automatic groups

We generalize the notion of a graph automatic group introduced by Kharlampovich, Khoussainov and Miasnikov (arXiv:1107.3645) by replacing the regular languages in their definition with more powerful language classes. For a fixed language class $\mathcal C$, we call the resulting groups $\mathcal C$-graph automatic. We prove that the class of $\mathcal C$-graph automatic groups is closed under change of generating set, direct and free product for certain classes $\mathcal C$. 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 $V$.

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arXiv2013

Some non-contracting automata groups

We add to the classification of groups generated by 3-state automata over a 2 letter alphabet given by Bondarenko et al., by showing that a number of the groups in the classification are non-contracting. We show that the criterion we use to prove a self-similar action is non-contracting also implies that the associated self-similarity graph introduced by Nekrashevych is non-hyperbolic.

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arXiv2013

Logspace computations for Garside groups of spindle type

M. Picantin introduced the notion of Garside groups of spindle type, generalizing the 3-strand braid group. We show that, for linear Garside groups of spindle type, a normal form and a solution to the conjugacy problem are logspace computable. For linear Garside groups of spindle type with homogenous presentation we compute a geodesic normal form in logspace.

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arXiv2013

Totally disconnected groups from Baumslag-Solitar groups

For each Baumslag-Solitar group BS(m,n) (m,n nonzero integers), a totally disconnected, locally compact group, G_{m,n}, is constructed so that BS(m,n) is identified with a dense subgroup of G_{m,n}. The scale function on G_{m,n}, a structural invariant for the topological group, is seen to distinguish the parameters m and n to the extent that the set of scale values is {(lcm(m,n)/|m|)^ρ, (lcm(m,n)/|n|)^ρ | ρ\in N}. It is also shown that G_{m,n} has flat rank 1 when |m|\neq |n| and 0 otherwise, and that G_{m,n} has a compact, open subgroup isomorphic to the product {(Z_p,+) | p is a prime divisor of the scale}.

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arXiv2013

On groups that have normal forms computable in logspace

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 finite extensions, finite index subgroups, direct products, wreath products, and also certain free products, 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.

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arXiv2012

On groups whose geodesic growth is polynomial

This note 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).

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arXiv2010

Some geodesic problems in groups

We consider several algorithmic problems concerning geodesics in finitely generated groups. We show that the three geodesic problems considered by Miasnikov et al [arXiv:0807.1032] are polynomial-time reducible to each other. We study two new geodesic problems which arise in a previous paper of the authors and Fusy [arXiv:0902.0202] .

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arXiv2010

Counting elements and geodesics in Thompson's group $F$

We present two quite different algorithms to compute the number of elements in the sphere of radius $n$ of Thompson's group $F$ with standard generating set. The first of these requires exponential time and polynomial space, but additionally computes the number of geodesics and is generalisable to many other groups. The second algorithm requires polynomial time and space and allows us to compute the size of the spheres of radius $n$ with $n \leq 1500$. Using the resulting series data we find that the growth rate of the group is bounded above by $2.62167...$. This is very close to Guba's lower bound of $\tfrac{3+\sqrt{5}}{2}$ \cite{Guba2004}. Indeed, numerical analysis of the series data strongly suggests that the growth rate of the group is exactly $\tfrac{3+\sqrt{5}}{2}$.

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arXiv2009

Random subgroups of Thompson's group $F$

We consider random subgroups of Thompson's group $F$ with respect to two natural stratifications of the set of all $k$ generator subgroups. We find that the isomorphism classes of subgroups which occur with positive density are not the same for the two stratifications. We give the first known examples of {\em persistent} subgroups, whose isomorphism classes occur with positive density within the set of $k$-generator subgroups, for all sufficiently large $k$. Additionally, Thompson's group provides the first example of a group without a generic isomorphism class of subgroup. Elements of $F$ are represented uniquely by reduced pairs of finite 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-finite and not algebraic. We then use the asymptotic growth to prove our density results.

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arXiv2008

An isoperimetric function for Stallings' group

We prove that $n^{7/3}$ is an isoperimetric function for a group of Stallings that is finitely presented but not of type $\mathcal{F}_3$. Note: The authors with Robert Young have now proved a quadratic Dehn function for this group. See arXiv:0712.3877

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arXiv2008

The Dehn function of Stallings' group

We prove that the Dehn function of a group of Stallings that is finitely presented but not of type F_3 is quadratic. To appear in Geometric and Functional Analysis.

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arXiv2006

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 recognised 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.

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arXiv2006

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 $\Z^n$ is accepted by a nondeterministic $m$-counter automaton if and only if $m \geq n$.

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arXiv2005

G-automata, counter languages and the Chomsky hierarchy

We consider how the languages of $G$-automata compare with other formal language classes. We prove that if the word problem of a group $G$ is accepted by a machine in the class $\mathcal M$ then the language of any $G$-automaton is in the class $\mathcal M$. It follows that the so called {\emph counter languages} (languages of $\mathbb Z^n$-automata) are context-sensitive, and further that counter languages are indexed if and only if the word problem for $\mathbb Z^n$ is indexed.

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