Maximally Symmetric Trees

We characterize the best model geometries for the class of virtually free groups, and we show that there is a countable infinity of distinct best model geometries in an appropriate sense – these are the maximally symmetric trees. The first theorem gives several equivalent conditions on a bounded valence, cocompact tree T without valence 1 vertices saying that T is maximally symmetric. The second theorem gives general constructions for maximally symmetric trees, showing for instance that every virtually free group has a maximally symmetric tree for a model geometry.     

Rank Rigidity for Cat(0) Cube Complexes

We prove that any group acting essentially without a fixed point at infinity on an irreducible finite-dimensional CAT(0) cube complex contains a rankone isometry. This implies that the Rank Rigidity Conjecture holds for CAT(0) cube complexes. We derive a number of other consequences for CAT(0) cube complexes, including a purely geometric proof of the Tits alternative, an existence result for regular elements in (possibly non-uniform) lattices acting on cube complexes, and a characterization of products of trees in terms of bounded cohomology.     

JSJ-splittings for finitely presented groups over slender groups

We generalize the JSJ-splitting of Rips and Sela to give decompositions of finitely presented groups which capture splittings over certain classes of small subgroups. Such classes include the class of all 2-ended groups and the class of all virtually Z⊕Z groups. The approach, called “track zipping”, is relatively elementary, and differs from the Rips-Sela approach in that it does not rely on the theory of R-trees but rather on an understanding of certain embedded 1-complexes (called patterns) in a presentation 2-complex for the ambient group. Oblatum 18-IV-1997 & 30-I-1998 / Published online: 18 September 1998     

Rates of change in high temperature electrical resistivity and oxygen diffusion coefficient in Ba2YCu3Ox

The change in electrical resistance with time for bulk, thick-film, and thin-film Ba₂YCu₃O_x at atmospheric pressure is described as a function of the oxygen partial pressure (100 to 0.001%) and temperature (320°–750°C). The potential usefulness of these materials as oxygen sensors is demonstrated. The rate of equilibration is faster during oxygen uptake than during its loss. Time constants to reach equilibration (1/e remaining), qualitatively scale with sample dimensions. For a 1m film at 600° C, <1 s for the range of P_O2 (O2 being a shorthand for O₂) from 100% to 0.001%. The rate increases markedly with increasing P_O2. The actual resistance decreases with P_O2 at a rate of log/log P_O2 = 0.4 at 700° C showing adequate sensitivity for sensor purposes. Times for the transient resistance change in the sample where used to estimate the oxygen diffusion coefficient in the ceramic. The diffusivities obtained are 4·10^–11–1·10^–12 cm²/s in the 435°–320° C range, with an activation energy of 27 kcal/mole.     

Splittings of certain Fuchsian groups

We provide an explicit splitting of Fuchsian groups of type as free products with amalgamation.

     

Widths of Subgroups

We say that the width of an infinite subgroup in is if there exists a collection of essentially distinct conjugates of such that the intersection of any two elements of the collection is infinite and is maximal possible. We define the width of a finite subgroup to be . We prove that a quasiconvex subgroup of a negatively curved group has finite width. It follows that geometrically finite surfaces in closed hyperbolic -manifolds satisfy the -plane property for some .

     

The Tits Alternative for Cat(0) Cubical Complexes

A Tits alternative theorem is proved in this paper for groupsacting on CAT(0) cubical complexes. That is, a proof is givento show that if G is assumed to be a group for which there isa bound on the orders of its finite subgroups, and if G actsproperly on a finite-dimensional CAT(0) cubical complex, theneither G contains a free subgroup of rank 2, or G is finitelygenerated and virtually abelian. In particular, the above conclusionholds for any group G with a free action on a finite-dimensionalCAT(0) cubical complex. 2000 Mathematics Subject Classification20F67, 20E08.