Eric L. Swenson

Recognizing constant curvature discrete groups in dimension 3

We characterize those discrete groups G which can act properly discontinuously, isometrically, and cocompactly on hyperbolic 3-space IHI3 in terms of the combinatorics of the action of G on its space at infinity. The major ingredients in the proof are the properties of groups that are negatively curved (in the large) (that is, Gromov hyperbolic), the combinatorial Riemann mapping theorem, and the Sullivan-Tukia theorem on groups which act uniformly quasiconformally on the 2-sphere.

Gene flow estimates in Utah's cougars imply management beyond Utah

We present results from a study of genetic variation in Utahs cougar population. Estimates were based on data for 50 animals at nine microsatellite loci with five individuals sampled for each of ten management units throughout Utah. Levels of variation were moderate (average genetic diversity across populations was estimated to be 0.4687 for all 50 individuals), and comparable with other large mammals. But this level of variation for the microsatellite loci translated into an inbreeding effective population size of only 571 animals, much lower than the current estimates of census sizes of around 2000-3000. A lack of differentiation among the sampled populations across Utah (average N e m = 6.2) indicates that gene flow occurs over a large area. Since cougars are capable of movement beyond the Utah state borders (and certainly across management units), a better understanding of migration rates and patterns of dispersal will be achieved by sampling a much larger geographic region incorporating much of the western USA. Successful management and conservation of this species will then require a far more integrated approach, involving agencies across a number of states, as opposed to current management practices involving individual units within states.

Quasi-convex groups of isometries of negatively curved spaces☆

Abstract Let H be a properly discontinuous group of isometries of a negatively curved (Gromov hyperbolic) metric space X . We give equivalent conditions on H to be quasi-convex. The main application of this is to give alternate definitions of quasi-convex, or rational subgroups of negatively curved (word hyperbolic) groups.

Hyperbolic elements in negatively curved groups

We show that in a negatively curved groupG the conjugacy class of any infinite cyclic subgroup contains a straight element, an elementg with |gn|=n|g|, and thus the translation number of an element in a negatively curved group is rational with uniformly bounded denominator. We also find an upper bound on the cardinality of a finite normal subgroup.

From continua to ℝ–trees

We show how to associate an R-tree to the set of cut points of a continuum. If X is a continuum without cut points we show how to associate an R-tree to the set of cut pairs of X.

Boundary dimension in negatively curved spaces

LetX be a negatively curved (Gromov hyperbolic) space. We construct a bound on dim ∂X when a group of isometries acts cocompactly onX. We construct an example of a negatively curved space with infinite-dimensional boundary.

On cyclic CAT(0) domains of discontinuity

Let

Tits rigidity of CAT(0) group boundaries

X

The algebraic torus theorem

be a CAT(0) space, and

Boundaries and JSJ Decompositions of CAT(0)-Groups

G