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In this thesis we study Kähler groups and their connections to Geometric Group Theory. This work presents substantial progress on three central questions in the field: (1) Which subgroups of direct products of surface groups are Kähler? (2) Which Kähler groups admit a classifying space with finite (n-1)-skeleton but no classifying space with finitely many n-cells? (3) Is it possible to give explicit finite presentations for any of the groups constructed in response to Question 2? Question 1 was raised by Delzant and Gromov. Question 2 is intimately related to Question 1: the non-trivial examples of Kähler subgroups of direct products of surface groups never admit a classifying space with finite skeleton. The only known source of non-trivial examples for Questions 1 and 2 are fundamental groups of fibres of holomorphic maps from a direct product of closed surfaces onto an elliptic curve; the first such construction is due to Dimca, Papadima and Suciu. Question 3 was posed by Suciu in the context of these examples. In this thesis we: provide the first constraints on Kähler subdirect products of surface groups (Theorem 7.3.1); develop new construction methods for Kähler groups from maps onto higher-dimensional complex tori (Section 6.1); apply these methods to obtain irreducible examples of Kähler subgroups of direct products of surface groups which arise from maps onto higher-dimensional tori and use them to show that our conditions in Theorem 7.3.1 are minimal (Theorem A); apply our construction methods to produce irreducible examples of Kähler groups that (i) have a classifying space with finite (n-1)-skeleton but no classifying space with finite n-skeleton and (ii) do not have a subgroup of finite index which embeds in a direct product of surface groups (Theorem 8.3.1); provide a new proof of Biswas, Mj and Pancholi's generalisation of Dimca, Papadima and Suciu's construction to more general maps onto elliptic curves (Theorem 4.3.2) and introduce invariants that distinguish many of the groups obtained from this construction (Theorem 4.6.2); and, construct explicit finite presentations for Dimca, Papadima and Suciu's groups thereby answering Question 3 (Theorem 5.4.4)).

A group Γ is of type Fn for some n ≥ 1 if it has a classifying complex with finite n-skeleton. These properties generalise the classical notions of finite generation and finite presentability. We investigate the higher finiteness properties for fibre products of groups.

For various values of n and m we investigate homomorphisms from Out(F_n) to Out(F_m) and from Out(F_n) to GL_m(K), i.e. the free and linear representations of Out(F_n) respectively. By means of a series of arguments revolving around the representation theory of finite symmetric subgroups of Out(F_n) we prove that each homomorphism from Out(F_n) to GL_m(K) factors through the natural map p_n from Out(F_n) to GL(H_1(F_n,Z)) = GL_n(Z) whenever n=3, m < 7 and char(K) is not an element of {2,3}, and whenever n>5, m< n(n+1)/2 and char(K) is not an element of {2,3,...,n+1}. We also construct a new infinite family of linear representations of Out(F_n) (where n > 2), which do not factor through p_n. When n is odd these have the smallest dimension among all known representations of Out(F_n) with this property. Using the above results we establish that the image of every homomorphism from Out(F_n) to Out(F_m) is finite whenever n=3 and n < m < 6, and of cardinality at most 2 whenever n > 5 and n < m < n(n-1)/2. We further show that the image is finite when n(n-1)/2 -1 < m < n(n+1)/2. We also consider the structure of normal finite index subgroups of Out(F_n). If N is such then we prove that if the derived subgroup of the intersection of N with the Torelli subgroup T_n < Out(F_n) contains some term of the lower central series of T_n then the abelianisation of N is finite.

The objects of study in this thesis are automorphism groups of free and right-angled Artin groups. Right-angled Artin groups are defined by a presentation where the only relations are commutators of the generating elements. When there are no relations the right-angled-Artin group is a free group and if we take all possible relations we have a free abelian group. We show that if no finite index subgroup of a group $G$ contains a normal subgroup that maps onto $mathbb{Z}$, then every homomorphism from $G$ to the outer automorphism group of a free group has finite image. The above criterion is satisfied by SL$_m(mathbb{Z})$ for $m geq 3$ and, more generally, all irreducible lattices in higher-rank, semisimple Lie groups with finite centre. Given a right-angled Artin group $A_Gamma$ we find an integer $n$, which may be easily read off from the presentation of $A_G$, such that if $m geq 3$ then SL$_m(mathbb{Z})$ is a subgroup of the outer automorphism group of $A_Gamma$ if and only if $m leq n$. More generally, we find criteria to prevent a group from having a homomorphism to the outer automorphism group of $A_Gamma$ with infinite image, and apply this to a large number of irreducible lattices as above. We study the subgroup $IA(A_Gamma)$ of $Aut(A_Gamma)$ that acts trivially on the abelianisation of $A_Gamma$. We show that $IA(A_Gamma)$ is residually torsion-free nilpotent and describe its abelianisation. This is complemented by a survey of previous results concerning the lower central series of $A_Gamma$. One of the commonly used generating sets of $Aut(F_n)$ is the set of Whitehead automorphisms. We describe a geometric method for decomposing an element of $Aut(F_n)$ as a product of Whitehead automorphisms via Stallings' folds. We finish with a brief discussion of the action of $Out(F_n)$ on Culler and Vogtmann's Outer Space. In particular we describe translation lengths of elements with regards to the `non-symmetric Lipschitz metric' on Outer Space.

This thesis proposes two new families of counter examples to the celebrated relation gap problem: firstly certain finite-index subgroups of right-angled Artin groups, and secondly groups with torsion arising as free products of HNN extensions of finite abelian groups. We prove a number of results relating the size of minimal presentations of the Artin subgroups to topological conditions on the associated flag complexes. We construct explicit presentations of these groups that provide considerable insight into when and why large families of relations are needed to present these groups, and cumulative evidence that these subgroups are extremely strong candidates for groups with relation gaps. For the examples with torsion, we prove that our groups have presentations whose associated relation modules can be generated by one fewer element than one expects. We prove that any presentation with this number of relations must be of a peculiar form. We also show that a criterion ofM. Lustig provides a non-trivial obstruction to the existence of relation gaps, but that for the presentations we are studying, this obstruction never arises. Finally, we give an explicit description of a finite 3-complex that is a counterexample to the D(2) onjecture if our groups have relation gaps.