3-manifolds with Heegaard Genus at Most Two

Acknowledgements My foremost thanks are to my supervisor Roman Nedela, for his continual encouragement, support and patience, for many inspirational conversations, leading me in the times of hasitations and careful reading many versions of the draft. Thanks also to Peter Maličk´y for the effort with some computations with groups and many advices in topology. Thanks also to Mathematical Institute and Agency for Support of Science and Technology (APVT) for the possibility to complete the study and the thesis. Especially, I would like to thank to Vivien Easson for her agreement to append her text into the thesis, which enabled me to shorten the Preface and helped me to understand recent progress in the geometrisation of 3-manifolds of genus two. I would like to thank to Sergei Matveev for many other informations about the geometrisation of 3-manifolds of genus two. Thanks also to Carlo Gagliardi, Paola Bandieri and Michele Mulazzani for pleasant and valuable stay on their universities, where the thesis got the final form. Thanks also to Gareth Jones, George Havas and Marston Conder for their advices concerning group presentations. Thanks also to Guido van Rossum et al. for developing interesting and powerful computer language, Python, in which my programs were written. My final thanks are to Lucia Cachovanová for continual understanding and support in the time of writing up and exploring a huge amout of mistypes in the draft.