Article

A Hurewicz theorem for the Assouad-Nagata dimension

06/2006; DOI: 10.1112/jlms/jdn005
Source: arXiv

ABSTRACT Given a function f : X → Y of metric spaces, the classical Hurewicz theorem states that dim(X) ≤ dim(f) + dim(Y). We provide analogs of this theorem for the Assouad–Nagata dimension, asymptotic Assouad–Nagata dimension, and asymptotic dimension (the latter result generalizes a theorem of Bell and Dranishnikov). As an application, we estimate the asymptotic Assouad–Nagata dimension of a finitely generated group G in terms of the asymptotic Assouad–Nagata dimensions of the groups K and H from the exact sequence 1 → K → G → H → 1.

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    ABSTRACT: ACKNOWLEDGMENTS I wish to thank my adviser, Alexander Dranishnikov, for his support, numerous enlightening discussions, and his patience while I was struggling. I would also like to thank the Department of Mathematics for the supportive atmosphere they have provided. I am also grateful to the College of Liberal Arts and Sciences for honoring me with the CLAS Dissertation Fellowship in my final semester. 4 TABLE OF CONTENTS page ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .,4 ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .,6 CHAPTER 1,INTRODUCTION AND BASIC DEFINITIONS . . . . . . . . . . . . . . . . . .,7 2,THE GRIGORCHUK GROUP . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3,COUNTABLE GROUPS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 4,THE SUBLINEAR COARSE STRUCTURE . . . . . . . . . . . . . . . . . . . . 22 5,THE DIMENSION OF THE SUBLINEAR HIGSON CORONA . . . . . . . . . 29 REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 BIOGRAPHICAL SKETCH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 5 Abstract of Dissertation Presented to the Graduate School

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