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 (<strong>Theorem 7.3.1</strong>); develop new construction methods for Kähler groups from maps onto higher-dimensional complex tori (<strong>Section 6.1</strong>); 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 (<strong>Theorem A</strong>); 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 (<strong>Theorem 8.3.1</strong>); 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 (<strong>Theorem 4.3.2</strong>) and introduce invariants that distinguish many of the groups obtained from this construction (<strong>Theorem 4.6.2</strong>); and, construct explicit finite presentations for Dimca, Papadima and Suciu's groups thereby answering Question 3 (<strong>Theorem 5.4.4)</strong>).
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:729135 |
| Date | January 2017 |
| Creators | Isenrich, Claudio Llosa |
| Contributors | Bridson, Martin R. |
| Publisher | University of Oxford |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
| Source | https://ora.ox.ac.uk/objects/uuid:4a7ab097-4de5-4b72-8fd6-41ff8861ffae |
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