The main work of this thesis starts with Chapter 2. Chapter 2 concerns the second homotopy module of a finitely presented group of type F3. We define the second higher order Dehn functions by considering the comparison between the "volume" and the "surface area" of nullhomotopies of spherical maps into CW complexes. We show that the second order Dehn function of groups of type F3 is an invariant of quasi-isometry type. In Chapter 3, we translate the concept of the second order Dehn function of finite group presentations to FDT monoid presentations by introducing a well-placed retraction relation between any two two-complexes and showing some invariance results. We show that the second order Dehn function of an FDT monoid at a fixed element is an invariant of isomorphism type. In Chapters 4, we give upper bounds for asynchronously combable groups with departure function. In Chapter 5, we first give the general upper bounds for direct products. Then we concentrate on the calculations for the optimal bounds of second order Dehn functions of direct products of the form G0 x F where the second order Dehn function of G0 is bounded by a linear function and F is a free group of finite rank. Some interesting examples are given. In Chapters 6, we carry out calculations for the upper bounds of second order Dehn functions of H N N-extensions, amalgamated free products, and split extensions, and finally in Chapter 7, some nice upper bounds as well as lower bounds for the second order Dehn functions of groups of the form Z2 x &phis;F are established where F is a free group of finite rank.
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Second Order Dehn Functions of Finitely Presented Groups and Monoids
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