This work examines gated chamber complexes, which can be thought of as generalisations of buildings in the sense of Tits, hyperplane arrangements which behave similar to reflection arrangements and it approaches the isomorphism problem for Coxeter groups.In the first part we develop some results for gated chamber complexes, which are motivated by the fact that Coxeter complexes and buildings also satisfy the property of being gated; i. e. all residues are gated sets. Our work continues some of the results in Mühlherrs dissertation, who examined spherical complexes in particular. We describe a reducibility type for firm gated chamber complexes and prove that segments in such complexes are already convex. Furthermore we generalise a technical result from the work of Mühlherr, Petersson and Weiss about descent in buildings to gated chamber complexes admitting an apartment system.The second part describes the foundations to handle hyperplane arrangements which decompose an open convex cone simplicially. This concept arises implicitly in some references in the literature, but we want to present it in a self-contained form. This yields the foundation for the correspondence between such arrangements, which in addition admit a crystallographic root system, and Cartan graphs admitting a real root system. This correspondence has already been shown for spherical arrangements by Cuntz. While it is not too hard to assign a Cartan graph to a crystallographic hyperplane arrangement, the other direction is more demanding. In particular it involves the reconstruction of the open convex cone from the given Cartan graph.Furthermore we describe some standard constructions, which are well known in the case of finite hyperplane arrangements. In particular, we describe subarrangements and induced arrangements.In the third part we study the isomorphism problem for Coxeter groups. For this decision problem Mühlherr conjectured a complete solution, the twist conjecture. In the right-angled case, Mühlherr used the reduction of distances of a Coxeter generating set within the Cayley graph to prove the conjecture. The conjecture has also been proven in several further cases. These cases either exclude Twists of a rank 2 or greater, or circumvent these by other methods.We were able to show the Twist conjecture for a new class of Coxeter groups, which do not contain certain rank 3 subdiagrams, including the rank 3 spherical and affine ones. With these technical conditions we are able to control rotation twists, which behave critically when reducing distances.
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