The growth series of H(ℤ × ℤ2)

Denote the Baumslag–Solitar family of groups as [Formula: see text]). When [Formula: see text] we study the Bass–Serre tree [Formula: see text] for [Formula: see text] as a geometric object. We suggest that the irregularity of [Formula: see text] is the principal obstruction for computing the growth series for the group. In the particular case [Formula: see text] we exhibit a set [Formula: see text] of normal form words having minimal length for [Formula: see text] and use it to derive various counting algorithms. The language [Formula: see text] is context-sensitive but not context-free. The tree [Formula: see text] has a self-similar structure and contains infinitely many cone types. All cones have the same asymptotic growth rate as [Formula: see text] itself. We derive bounds for this growth rate, the lower bound also being a bound on the growth rate of [Formula: see text].

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Growth in Baumslag–Solitar groups I: subgroups and rationality

LMS Journal of Computation and Mathematics ◽

10.1112/s146115700900028x ◽

2011 ◽

Vol 14 ◽

pp. 34-71 ◽

Cited By ~ 4

Author(s):

Eric M. Freden ◽

Teresa Knudson ◽

Jennifer Schofield

Keyword(s):

Turing Machine ◽

Time Complexity ◽

Machine Construction ◽

Convolution Product ◽

Growth Series ◽

Context Sensitive ◽

Context Free Language ◽

Time And Space Complexity ◽

Context Free ◽

Free Language

AbstractThe computation of growth series for the higher Baumslag–Solitar groups is an open problem first posed by de la Harpe and Grigorchuk. We study the growth of the horocyclic subgroup as the key to the overall growth of these Baumslag–Solitar groups BS(p,q), where 1<p<q. In fact, the overall growth series can be represented as a modified convolution product with one of the factors being based on the series for the horocyclic subgroup. We exhibit two distinct algorithms that compute the growth of the horocyclic subgroup and discuss the time and space complexity of these algorithms. We show that when p divides q, the horocyclic subgroup has a geodesic combing whose words form a context-free (in fact, one-counter) language. A theorem of Chomsky–Schützenberger allows us to compute the growth series for this subgroup, which is rational. When p does not divide q, we show that no geodesic combing for the horocyclic subgroup forms a context-free language, although there is a context-sensitive geodesic combing. We exhibit a specific linearly bounded Turing machine that accepts this language (with quadratic time complexity) in the case of BS(2,3) and outline the Turing machine construction in the general case.

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Ontogeny of Drevermannia and the origin of blindness in Late Devonian proetoid trilobites

Geological Magazine ◽

10.1017/s0016756805001421 ◽

2006 ◽

Vol 143 (1) ◽

pp. 89-104 ◽

Cited By ~ 10

Author(s):

RUDY LEROSEY-AUBRIL

Keyword(s):

Ontogenetic Development ◽

Late Devonian ◽

Developmental Strategy ◽

Larval Size ◽

Growth Series ◽

Extinction Event ◽

Alternative Hypotheses ◽

Devonian Extinction

Numerous silicified and calcareous sclerites of various sizes, recovered from the latest Famennian of Thuringia (Germany), allow the description of the first complete growth series of a blind proetoid trilobite: Drevermannia richteri. In addition, the partial ontogenetic development of Drevermannia antecurvata sp. nov. and undetermined species, Drevermannia sp. 1, are described. The proetoid anaprotaspides, associated with D. richteri, illustrate that a marked increase in larval size occurred prior to the terminal Devonian extinction event. Considering the homogeneity of larval size in older Devonian proetoids, it is interpreted as evidence that the developmental strategy of these trilobites was significantly modified. Though largely speculative, two alternative hypotheses are proposed to explain this modification. Finally, all three ontogenetic sequences show that ocular structures never develop externally in Drevermannia, but also illustrate that the development of optical nerves is not completely lost in this group. This suggests that blindness in the Drevermannia lineage followed a centripetal mode of eye reduction.

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