Recognition of subgroups of direct products of hyperbolic groups

We give examples of direct products of three hyperbolic groups in which there cannot exist an algorithm to decide which finitely presented subgroups are isomorphic.

     

Synthesis of Calix

The formation of p-tert-butylcalix[4]arene mono-, bis-, tris-, and tetrakistriflates 3, 2, 4, and 5, respectively, and their respective reactions, under typical Pd-catalyzed carbonylative, Suzuki-Miyaura coupling, or deoxygenation conditions are described. A novel, nonsolvent-derived 1:1 clathrate (6) of benzophenone and 3 was formed from the palladium-catalyzed carbonylative reaction of phenylboronic acid and 2. The X-ray crystal structure of this first nonsolvent-derived clathrate of a calix[4]arene derivative is reported. Another 1:1 clathrate of triethylamine and 3 was formed during the attempted Pd-catalyzed deoxygenation of 2.     

Polynomial Dehn Functions and the Length of Asynchronously Automatic Structures

We extend the range of observed behaviour among length functionsof optimal asynchronously automatic structures. We do so bymeans of a construction that yields asynchronously automaticgroups with finite aspherical presentations where the Dehn functionof the group is polynomial of arbitrary degree. Many of thesegroups can be embedded in the automorphism group of a free group.Moreover, the fact that the groups have aspherical presentationsmakes them useful tools in the search to determine the spectrumof exponents for second order Dehn functions. We contributeto this search by giving the first exact calculations of groupswith quadratic and superquadratic exponents. 2000 Mathematical Subject Classification: 20F06, 20F65, 20F69.     

X-ray structure of 1,3-adamantane dicarbonamide

The X-ray crystal structure of 1,3-adamantane-dicarbonamide has been determined. Colourless, cube shaped crystals of C₁₂H₁₈N₂O₂ crystallize in the monoclinic space group C2/c (#15) with cell dimensionsa=7·934(3),b=15·543(3),c=9·404(2) Å, =93.63°;V=1157.4(5) ų andZ=4.1074 unique reflections withI _net >2.00(I) on refinement afforded values ofR=0.039 andR _w =0.040.     

On the Geometry of the Automorphism Group of a Free Group

The groups Aut(F₃) and Out(F₃) satisfy strictly exponentialisoperimetric inequalities; in particular, they are not automatic.For n 3, Aut (F_n) and Out (F_n) do not admit bounded bicombingsof sub-exponential length, hence they cannot act properly andcocompactly by isometries on any simply-connected space of non-positivecurvature, and they are not biautomatic.     

Fibre products, non-positive curvature, and decision problems

We give a criterion for fibre products to be finitely presented and use it as the basis of a construction that encodes the pathologies of finite group presentations into pairs of groups where G is a product of hyperbolic groups and P is a finitely presented subgroup. This enables us to prove that there is a finitely presented subgroup P in a biautomatic group G such that the generalized word problem for is unsolvable and P has an unsolvable conjugacy problem. An additional construction shows that there exists a compact non-positively curved polyhedron X such that is biautomatic and there is no algorithm to decide isomorphism among the finitely presented subgroups of . Received: October 7, 1999.     

Fractional isoperimetric inequalities and subgroup distortion

It is shown that there exist infinitely many non-integers such that the Dehn function of some finitely presented group is . Explicit examples of such groups are constructed. For each rational number pairs of finitely presented groups are constructed so that the distortion of in is .

     

Isoperimetric inequalities for the fundamental groups of torus bundles over the circle

We give upper bounds for isoperimetric functions of semi-direct products in terms of the asymptotic behaviour of ||A ^k || ask . In the caseA Sp(n, ) we show that these bounds are sharp. This enables us to describe infinite families of nilpotent groups whose Dehn functions are bounded above and below by polynomials of degree the nilpotency class plus 1. We also recover the isoperimetric inequalities of cocompact lattices inNil andSol.Partially supported by NSF grant DMS-9203500.