ON QUASICONVEX SUBSETS OF HYPERBOLIC GROUPS By 2005

Abstract:  Dedicated to my mother, Margarita, and my brother, Vardan. ii ACKNOWLEDGMENTS I would like to express my gratitude to the people who helped me on the way to writing this thesis. First of all, I thank my family for the care and support you provided for all these years. Your encouragement is the very reason why this work exists. I am deeply indebted to my advisor, Professor Alexander Ol?shanskii, for his guidance, help and patience during my studies at the Moscow State University and Vanderbilt. Doctor Ol?shanskii has been directing my research from the very first steps; the important lessons he taught will remain with me for the rest of my life. I would like to thank Professors Mark Sapir and Mike Mihalik for all the helpful discus-sions we had and the extensive amount of information I learned from them. I had a privilege to attend a lot of interesting courses taught by Vanderbilt Professors Bruce Hughes, Ralph McKenzie and Guoliang Yu which had a significant impact on my research. I am very grateful to these people. I am also thankful to Doctor Tom Kephart from Vanderbilt Physics Department for being a member of my Ph.D. committee. Throughout the five years I spent at Vanderbilt, I was honored to work with fellow graduate students Dmitriy Sonkin, Alexey Muranov, Maxym Yattselev, Erwin Mi?a-Diaz,