Free group outer automorphisms were shown by Bestvina and Randell to have fixed subgroups whose rank is bounded in terms of the rank of the underlying group. We consider the case where this upper bound is achieved and obtain combinatorial results about such outer automorphisms thus extending the work of Collins and Turner. We go on to show that such automorphisms can be represented by certain graph of group isomorphisms called Dehn Twists and also solve the conjuagacy problem in a restricted case, thus reproducing the work of Cohen and Lustig, but with different methods. We rely heavily on the relative train tracks of Bestvina and Randell and in fact go on to use an analogue of these for free product automorphisms developed by Collins and Turner. We prove an index theorem for such automorphisms which counts not only the group elements which are fixed but also the points which are fixed at infinity - the infinite reduced words. Two applications of this theorem are considered, first to irreducible free group automorphisms and then to the action of an automorphism on the boundary of a hyperbolic group. We reduce the problem of counting the number of points fixed on the. boundary to the case where the hyperbolic group is indecomposable and provide an easy application to virtually free groups.

In this thesis, we study the representation theory of the automorphism group Aut (Fn) of a free group by studying the representation theory of three finite subgroups: two symmetric groups, Sn and Sn+1, and a Coxeter group of type Bn, also known as a hyperoctahedral group. The representation theory of these subgroups is well understood in the language of Young Diagrams, and we apply this knowledge to better understand the representation theory of Aut (Fn). We also calculate irreducible representations of Aut (Fn) in low dimensions and for small n.

Given an action of a countable group with a quasi-invariant measure, there exists a multiplicative group in (0, ∞), called the ratio set of the group action, which in a sense describes the values of the Radon-Nikodym derivative. The main purpose of this thesis is to find the ratio set of the action of a finitely generated free group Ƒ on its topological boundary ∂Ƒ (the set of infinite words) for a certain natural class of quasi-invariant boundary measures. -- In Section 1, we focus on the general ergodic theory of equivalence relations. We outline the set-up, borrow from [1], [4] the definitions of the central notions of the theory, including counting measures (Proposition 1.8), quasi-invariance (Definition 1.6), Radon-Nikodym cocycle (Definition 1.15) and raio set (Definition 1.19), and illustrate them on the example of the orbit equivalence relation of a Markov shift (Definition 1.22). We also introduce the principal object: the boundary action of a finitely generated free group (see Section 1.2). -- In Section 2, we define the class of multiplicative Markov measures (Definition 2.1). These are the measures on a topological Markov chain entirely determined just by an initial (base) distribution and the admissibility matrix; the transition probabilities are then just the normalized restrictions of the base distribution onto the set of admissible transitions (see [7]). In the case of the free group, its boundary has a natural structure of a topological Markov chain (determined by the irreducibility condition from the definition of a free group: consecutive letters should not cancel each other), and in this case, we show that the multiplicative Markov measures are precisely the ones for which the Radon-Nikodym cocycle is a product cocycle (i.e. a cocycle whose potential only depends on the first letter of the input; see Definition 2.8). The final result of this section is an explicit description of the ratio set of the boundary action with respect to multiplicative Markov measures. -- In Section 3, given a probability measure 𝜇 on the set of free generators and their inverses, the definition of the associated nearest neighbor random walk is given. According to Furstenberg's Theorem (proof provided in Appendix), in this random walk, sample paths converge almost surely to a random boundary point, and the resulting limit distribution on the boundary of the free group is called the harmonic measure of the random walk (see Section 3.1). We show that the harmonic measure is a multiplicative measure (Theorem 3.3), and therefore the results of Section 2 allow us to describe the ratio set of the harmonic measure (Theorem 3.5). A significant role in these considerations is played by the passage probabilities of the random walk (given a group element, the probability that it is ever visited by a random walk). Since the harmonic measure is multiplicative, its potential only depends on the first letter, and this dependence actually amounts to taking the inverse of the corresponding passage probability (Proposition 2.9, Remark 2.10). Finally, we establish a one-to-one correspondence between three families of numbers indexed by the alphabet of the free group and subject to natural conditions; these are the step distributions of the random walk, the base of the harmonic measure (which is multiplicative Markov) and the family of passage probabilities (Theorem 3.6). -- In Section 4, we discuss another method for finding the ratio set of the harmonic measure based on using Martin theory (see [2]). -- In the Appendix, we prove Furstenberg's Theorem, a result used for defining the harmonic measure in Section 3. Actually, it is applicable not only for the nearest neighbor random walk (i.e. not only when the probability measure 𝜇 is supported on the alphabet set) but also the more general case where the support of the step distribution generates the free group. Moreover, in addition to the existence it also characterizes the harmonic measure as the unique 𝜇-stationary measure on the boundary

We investigate the genericity of measure-preserving actions of the free group Fn, on possibly countably infinitely many generators, acting on a standard probability space. Specifically, we endow the space of all measure-preserving actions of Fn acting on a standard probability space with the weak topology and explore what properties may be verified on a comeager set in this topology. In this setting we show an analog of the classical Rokhlin Lemma. From this result we conclude that every action of Fn may be approximated by actions which factor through a finite group. Using this finite approximation we show the actions of Fn, which are rigid and hence fail to be mixing, are generic. Combined with a recent result of Kerr and Li, we obtain that a generic action of Fn is weak mixing but not mixing. We also show a generic action of Fn has sigma-entropy at most zero. With some additional work, we show the finite approximation result may be used to that show for any action of Fn, the crossed product embeds into the tracial ultraproduct of the hyperfinite II1 factor. We conclude by showing the finite approximation result may be transferred to a subspace of the space of all topological actions of Fn on the Cantor set. Within this class, we show the set of actions with sigma-entropy at most zero is generic.

Este projeto de mestrado tem por objetivo dar uma prova elementar do seguinte teorema de Tits, conhecido como Teorema da Alternativa de Tits: Seja G um grupo linear finitamente gerado sobre um corpo. Então G é solúvel por finito ou G contém um grupo livre não cíclico. Este teorema, que foi provado por J. Tits em 1972 [4], foi considerado pelo matemático J.P. Serre como um dos mais importantes resultados de álgebra do século XX. Quando dizemos uma prova elementar, não queremos absolutamente te dizer uma prova simples. Seguiremos a prova simplificada de John D. Dixon / This masters project aims to give an elementary proof of the following theorem of Tits, known as the Alternative Tits Theorem: Let G be a finitely generated linear group over a field. Then either G is solvable by finite or G contains a noncyclic free subgroup. This theorem was proved by J. Tits in 1972 [4], was considered by the mathematician J.P. Serre, as one of the most important algebra results of the XX century. When we say an elementary proof, we absolutely not mean a simple proof. We will follow the simplified proof of John D. Dixon

<p>&nbsp / </p> <p align="left">The groups we consider in this study belong to the class <font face="F30">X</font><font face="F25" size="1"><font face="F25" size="1">0 </font></font><font face="F15">of all finitely generated groups with finite commutator subgroups.</font></p>

We examine the amalgamated free product of hyperfinite von Neumann algebras. First we describe the amalgamated free product of hyperfinite von Neumann algebras over finite dimensional subalgebras. In this case the result is always the direct sum of a hyperfinite von Neumann algebra and a finite number of interpolated free group factors. We then show that this class is closed under this type of amalgamated free product. After that we allow amalgamation over possibly infinite dimensional multimatrix subalgebras. In this case the product of two hyperfinite von Neumann algebras is the direct sum of a hyperfinite von Neumann algebra and a countable direct sum of interpolated free group factors. As before, we show that this class is closed under amalgamated free products over multimatrix algebras.

<p>&nbsp / </p> <p align="left">The groups we consider in this study belong to the class <font face="F30">X</font><font face="F25" size="1"><font face="F25" size="1">0 </font></font><font face="F15">of all finitely generated groups with finite commutator subgroups.</font></p>

Magister Scientiae - MSc / The groups we consider in this study belong to the class X0 of all nitely generated groups with nite commutator subgroups. We shall eventually narrow down to the groups of the form T owZn for some n 2 N and some nite abelian group T. For a X0-group H, we study the non-cancellation set, (H), which is de ned to be the set of all isomorphism classes of groups K such that H Z = K Z. For X0-groups H, on (H) there is an abelian group structure [38], de ned in terms of embeddings of K into H, for groups K of which the isomorphism classes belong to (H). If H is a nilpotent X0-group, then the group (H) is the same as the Hilton-Mislin (see [10]) genus group G(H) of H. A number of calculations of such Hilton-Mislin genus groups can be found in the literature, and in particular there is a very nice calculation in article [11] of Hilton and Scevenels. The main aim of this thesis is to compute non-cancellation (or genus) groups of special types of X0-groups such as mentioned above. The groups in question can in fact be considered to be direct products of metacyclic groups, very much as in [11]. We shall make extensive use of the methods developed in [30] and employ computer algebra packages to compute determinants of endomorphisms of nite groups. / South Africa

Este projeto de mestrado tem por objetivo dar uma prova elementar do seguinte teorema de Tits, conhecido como Teorema da Alternativa de Tits: Seja G um grupo linear finitamente gerado sobre um corpo. Então G é solúvel por finito ou G contém um grupo livre não cíclico. Este teorema, que foi provado por J. Tits em 1972 [4], foi considerado pelo matemático J.P. Serre como um dos mais importantes resultados de álgebra do século XX. Quando dizemos uma prova elementar, não queremos absolutamente te dizer uma prova simples. Seguiremos a prova simplificada de John D. Dixon / This masters project aims to give an elementary proof of the following theorem of Tits, known as the Alternative Tits Theorem: Let G be a finitely generated linear group over a field. Then either G is solvable by finite or G contains a noncyclic free subgroup. This theorem was proved by J. Tits in 1972 [4], was considered by the mathematician J.P. Serre, as one of the most important algebra results of the XX century. When we say an elementary proof, we absolutely not mean a simple proof. We will follow the simplified proof of John D. Dixon