Dehn filling in relatively hyperbolic groups

We introduce a number of new tools for the study of relatively hyperbolic groups. First, given a relatively hyperbolic group G, we construct a nice combinatorial Gromov hyperbolic model space acted on properly by G, which reflects the relative hyperbolicity of G in many natural ways. Second, we construct two useful bicombings on this space. The first of these, "preferred paths", is combinatorial in nature and allows us to define the second, a relatively hyperbolic version of a construction of Mineyev. As an application, we prove a group-theoretic analog of the Gromov-Thurston 2\pi Theorem in the context of relatively hyperbolic groups.Comment: 83 pages. v2: An improved version of preferred paths is given, in which preferred triangles no longer need feet. v3: Fixed several small errors pointed out by the referee, and repaired several broken figures. v4: corrected definition 2.38. This is very close to the published versio

An alternate proof of Wise's Malnormal Special Quotient Theorem

We give an alternate proof of Wise's Malnormal Special Quotient Theorem (MSQT), avoiding cubical small cancellation theory. We also show how to deduce Wise's Quasiconvex Hierarchy Theorem from the MSQT and theorems of Hsu--Wise and Haglund--Wise.Comment: 42 pages, 10 figures. Version 2 contains minor changes, addressing referee comments. To appear in Forum of Mathematics, P

Hyperbolic groups acting improperly

In this paper we study hyperbolic groups acting on CAT(0) cube complexes. The first main result (Theorem A) is a structural result about the Sageev construction, in which we relate quasi-convexity of hyperplane stabilizers with quasi-convexity of cell stabilizers. The second main result (Theorem D) generalizes both Agol's theorem on cubulated hyperbolic groups and Wise's Quasi-convex Hierarchy Theorem.Comment: 52pp. In v3, some unnecessary assumptions are dropped from some technical results, especially in Section 5 and Corollary 6.5. The main results are unchanged, but the improved technical results are expected to be useful in future work. Several other small improvements to the exposition have been mad

Residual finiteness, QCERF, and fillings of hyperbolic groups

We prove that if every hyperbolic group is residually finite, then every quasi-convex subgroup of every hyperbolic group is separable. The main tool is relatively hyperbolic Dehn filling.Comment: (v1) 22 pages, 2 figures. (v2) 24 pages, 2 figures. An error in the proof and statement of the main technical lemma was corrected, and some other small corrections and clarifications were mad

Multiple Storm Event Impacts on Epikarst Storage and Transport of Organic Soil Amendments in South-central Kentucky

The groundwater in agricultural karst areas, such as Kentucky’s Pennyroyal Plateau, is susceptible to contamination from organic soil amendments and pesticides. During 2011, water samples and geochemical data were collected every four hours before, during, and between storm events from a waterfall in Crumps Cave from January to September to track the transport and residence time of epikarst water and organic soil amendments during variable fl ow conditions. Geochemical data consisting of pH, specific conductivity, temperature, and discharge were collected continuously at 10-minute intervals, along with rainfall amounts. The changes in geochemistry indicate simultaneous storage and transport of meteoric water through epikarst pathways into the cave, with rapid transport of bacteria occurring through the conduits that bypass storage. Results indicate current best management practices in agricultural karst areas need to be revisited to incorporate areas that do not have surface runoff but where contaminants are transported by seepage into local aquifers