We studied the subjectivity of word maps on SU(n) and the length of the shortest elements in the central series of free group of rank 2 with some applications to almost laws in compact groups.

We studied the subjectivity of word maps on SU(n) and the length of the shortest elements in the central series of free group of rank 2 with some applications to almost laws in compact groups.

In this thesis I study metric aspects of finite nearly simple groups. Its four distinct chapters deal with four different questions. In the first chapter, I give a full description of the normal subgroup lattice of any algebraic ultraproduct of universal finite quasisimple groups. In the second, I investigate approximation questions for arbitrary abstract and topological groups by families of finite groups with conjugacy-invariant norms. In the third chapter, I prove that the map induced by any non-trivial word on the metric ultraproduct of classical groups of Lie type of unbounded rank is always surjective using cohomological and algebraic methods. In the last chapter, it is proved that (simple) metric ultraproducts of finite classical groups of Lie type of unbounded rank with different field sizes are always non-isomorphic. Also, if the field sizes are equal, two such ultraproducts can only be isomorphic if the Lie types are equal or one Lie type is orthogonal and the other symplectic.:Introduction 0 Notation, basic definitions, and facts 0.1 Group theory 0.2 Some ring and field theory 0.3 Ultraproducts and norms 1 The normal subgroup lattice of an algebraic ultraproduct 1.1 Introduction 1.2 Auxiliary geometric results 1.3 Relative bounded normal generation in universal finite quasisimple groups 1.4 The lattice of normal subgroups 2 Metric approximation of groups by finite groups 2.1 Introduction 2.2 Preliminaries 2.2.1 On C-approximable abstract groups 2.2.2 On C-approximable topological groups 2.3 On Sol-approximable groups 2.4 On Fin-approximable groups 2.5 On the approximability of Lie groups 3 Word maps are surjective on metric ultraproducts 3.1 Introduction 3.2 Symmetric groups 3.2.1 Power words 3.2.2 The cycle structure of elements from PSL_2(q) 3.2.3 Effective surjectivity of word maps over finite fields 3.2.4 Proof of Theorem 3.1 3.3 Unitary groups 3.3.1 Proof of Theorem 3.3 3.3.2 Further implications 3.3.3 Concluding remarks 3.4 Finite groups of Lie type 3.4.1 The linear case 3.4.2 The case of quasisimple groups of Lie type stabilizing a form 3.4.3 An alternative way of proving Theorem 3.1 using wreath products 4 Isomorphism questions for metric ultraproducts 4.1 Introduction 4.2 Description of conjugacy classes in S_U, GL_U(q), and PGL_U(q) 4.3 Characterization of torsion elements in S_U , GL_U(q), and PGL_U(q) 4.4 Faithful action of S_U and PGL_U(q) 4.5 Centralizers in S_U , GL_U(q), Sp_U(q), GO_U(q), and GU_U(q) 4.6 Centralizers in PGL_U(q), PSp_U(q), PGO_U(q), and PGU_U(q) 4.7 Double centralizers of torsion elements 4.7.1 The case S_U 4.7.2 The case PGL_U(q), PSp_U(q), PGO_U(q), and PGU_U(q) 4.8 Distinction of metric ultraproducts 4.8.1 Computation of e_G(o) when gcd{o,p}=gcd{o,|Z|}=1 4.8.2 Proof of Theorem 4.1 Index of Symbols Index Bibliography / In dieser Doktorarbeit studiere ich metrische Aspekte von endlichen fast-einfachen Gruppen. Ihre vier Kapitel beschäftigen sich mit vier unterschiedlichen Themenfeldern. Im ersten Kapitel gebe ich eine vollständige Beschreibung des Normalteilerverbandes eines algebraischen Ultraproduktes von universellen endlichen quasieinfachen Gruppen. Im zweiten beschäftige ich mich mit Approximationsfragen für beliebige abstrakte und topologische Gruppen durch Familien von endlichen Gruppen, auf denen eine konjugationsinvariante Norm erklärt ist. Im dritten Kapitel beweise ich, dass die Abbildung auf einem metrischen Ultraprodukt von klassischen Gruppen vom Lie-Typ von unbeschränktem Rang, die von einem beliebigen nicht-trivialen Wort induziert wird, immer surjektiv ist. Dabei verwende ich sowohl kohomologische als auch algebraische Methoden. Im letzten Kapitel beweise ich, dass (einfache) metrische Ultraprodukte von klassischen endlichen Gruppen vom Lie-Typ von unbeschränktem Rang mit unterschiedlicher Körpergröße immer nicht-isomorph sind. Ist die Körpergröße gleich, so können zwei solche Gruppen nur dann isomorph sein, falls sie auch denselben Lie-Typ haben, oder eine vom orthogonalen Typ und die andere vom symplektischen ist.:Introduction 0 Notation, basic definitions, and facts 0.1 Group theory 0.2 Some ring and field theory 0.3 Ultraproducts and norms 1 The normal subgroup lattice of an algebraic ultraproduct 1.1 Introduction 1.2 Auxiliary geometric results 1.3 Relative bounded normal generation in universal finite quasisimple groups 1.4 The lattice of normal subgroups 2 Metric approximation of groups by finite groups 2.1 Introduction 2.2 Preliminaries 2.2.1 On C-approximable abstract groups 2.2.2 On C-approximable topological groups 2.3 On Sol-approximable groups 2.4 On Fin-approximable groups 2.5 On the approximability of Lie groups 3 Word maps are surjective on metric ultraproducts 3.1 Introduction 3.2 Symmetric groups 3.2.1 Power words 3.2.2 The cycle structure of elements from PSL_2(q) 3.2.3 Effective surjectivity of word maps over finite fields 3.2.4 Proof of Theorem 3.1 3.3 Unitary groups 3.3.1 Proof of Theorem 3.3 3.3.2 Further implications 3.3.3 Concluding remarks 3.4 Finite groups of Lie type 3.4.1 The linear case 3.4.2 The case of quasisimple groups of Lie type stabilizing a form 3.4.3 An alternative way of proving Theorem 3.1 using wreath products 4 Isomorphism questions for metric ultraproducts 4.1 Introduction 4.2 Description of conjugacy classes in S_U, GL_U(q), and PGL_U(q) 4.3 Characterization of torsion elements in S_U , GL_U(q), and PGL_U(q) 4.4 Faithful action of S_U and PGL_U(q) 4.5 Centralizers in S_U , GL_U(q), Sp_U(q), GO_U(q), and GU_U(q) 4.6 Centralizers in PGL_U(q), PSp_U(q), PGO_U(q), and PGU_U(q) 4.7 Double centralizers of torsion elements 4.7.1 The case S_U 4.7.2 The case PGL_U(q), PSp_U(q), PGO_U(q), and PGU_U(q) 4.8 Distinction of metric ultraproducts 4.8.1 Computation of e_G(o) when gcd{o,p}=gcd{o,|Z|}=1 4.8.2 Proof of Theorem 4.1 Index of Symbols Index Bibliography