Title: Volume of Pyramid Author: Bc. Milan Vaňkát Department: Department of Mathematics Education Supervisor: Mgr. Zdeněk Halas, DiS., Ph.D. Abstract: The subject of this thesis is Hilbert's third problem. In the first chapter we follow it's roots back to Euclid's Elements. We focus in particular on the theorem that triangular pyramids of equal altitudes are to each other as their bases. We also discuss analogous statements for triangles, parallelograms and parallelepipeds. We point out the way in which the issues of content and volume of geometrical figures were approached in Greek mathematics. In the second chapter we present the historical background of Hilbert's third problem. We outline the development of methods of it's solution - from M. Dehn's first answer in 1901 to the abstract definition of Dehn invariants as a R ⊗Z Rπ- valued functional on the polyhedral group that was introduced by B. Jessen in 1968. Later we construct Dehn invariants and present a thorough solution to the Hilbert's third problem. In the end we sketch out mathematical issues connected to this problem that have been studied in the second half of 20th century. An illustrative high school exercise on derivation of the volume formula for py- ramid by Eudoxus's method of exhaustion is included in the appendix. Keywords: pyramid, volume,...

In 1911, mathematician Max Dehn posed three decision problems for finitely presented groups that have remained central to the study of combinatorial group theory. His work provided the foundation for geometric group theory, which aims to analyze groups using the topological and geometric properties of the spaces they act on. In this thesis, we study group actions on Cayley graphs and the Farey tree. We prove that a group has a solvable word problem if and only if its associated Cayley graph is constructible. Moreover, we prove that a group is finitely generated if and only if it acts geometrically on a proper path-connected metric space. As an example, we show that SL(2, Z) is finitely generated by proving that it acts geometrically on the Farey tree.

Given a presentation of G, the word problem asks whether there exists an algorithm to determine which words in the free group, F(A), represent the identity in G. In this thesis, we study small cancellation theory, developed by Lyndon, Schupp, and Greendlinger in the mid-1960s, which contributed to the resurgence of geometric group theory. We investigate the connection between Van Kampen diagrams and the small cancellation hypotheses. Groups that have a presentation satisfying the small cancellation hypotheses C'(1/6), or C'(1/4) and T(4) have a nice solution to the word problem known as Dehn’s Algorithm.