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Gray and Kambites introduced a notion of hyperbolicity in the setting of semimetric spaces like digraphs or semigroups. We carry over some of the fundamental results of hyperbolic spaces to this new setting. In particular, we prove under some small additional geometric assumption that their notion of hyperbolicity is preserved by quasi-isometries. We also construct a boundary based on quasi-geodesic rays and anti-rays that is preserved by quasi-isometries and, in the case of locally finite digraphs, refines their ends. We show that it is possible to equip the space, if it is finitely based, with its boundary with a pseudo-semimetric and show some further results for the boundary. We also apply our results to semigroups and give a partial solution to a problem of Gray and Kambites., 44 pages

In this note, we show that locally finite quasi-transitive graphs are quasi-isometric to trees if and only if every other locally finite quasi-transitive graph quasi-isometric to them is minor excluded. This generalizes results by Ostrovskii and Rosenthal and by Khukhro on minor exclusion for groups., Comment: 4 pages

We prove two characterisations of accessibility of locally finite quasi-transitive connected graphs. First, we prove that any such graph $G$ is accessible if and only if its set of separations of finite order is an ${\rm Aut}(G)$-finitely generated semiring. The second characterisation says that $G$ is accessible if and only if every process of splittings in terms of tree amalgamations stops after finitely many steps., 15 pages

We generalise the standard constructions of a Cayley graph in terms of a group presentation by allowing some vertices to obey different relators than others. The resulting notion of presentation allows us to represent every vertex transitive graph. As an intermediate step, we prove that every countably infinite, connected, vertex transitive graph has a perfect matching. Incidentally, we construct an example of a 2-ended cubic vertex transitive graph which is not a Cayley graph, answering a question of Watkins from 1990., Comment: 29 pages, 8 figures, main authors: Agelos Georgakopoulos and Alex Wendland, appendix by: Matthias Hamann and Alex Wendland

We look at tree amalgamations of locally finite quasi-transitive hyperbolic graphs and prove that the homeomorphism type of the hyperbolic boundary of such a tree amalgamation only depends on the homeomorphism types of the hyperbolic boundaries of their factors. Additionally, we show that two locally finite quasi-transitive hyperbolic graphs have homeomorphic hyperbolic boundaries if and only if the homeomorphism types of the hyperbolic boundaries of the factors of their terminal factorisations coincide., Comment: 9 pages

We prove the existence of an upper bound on the asymptotic dimension of tree amalgamations of locally finite quasi-transitive connected graphs. This generalises a result of Dranishnikov for free products with amalgamation and a result of Tselekidis for HNN-extensions of groups to tree amalgamations of graphs. As a corollary, we obtain an upper bound on the asymptotic dimension of a multi-ended quasi-transitive locally finite graph based on any of their factorisations., Comment: 7 pages

We investigate the connections between tree amalgamations and quasi-isometries. In particular, we prove that the quasi-isometry type of multi-ended accessible quasi-transitive connected locally finite graphs is determined by the quasi-isometry type of their one-ended factors in any of their terminal factorisations. Our results carry over theorems of Papsoglu and Whyte on quasi-isometries between multi-ended groups to those between multi-ended graphs. In the end, we discuss the impact of our results to a question of Woess., Comment: 14 pages

We prove that the cut space of any transitive graph $G$ is a finitely generated ${\rm Aut}(G)$-module if the same is true for its cycle space. This confirms a conjecture of Diestel which says that every locally finite transitive graph whose cycle space is generated by cycles of bounded length is accessible. In addition, it implies Dunwoody's conjecture that locally finite hyperbolic transitive graphs are accessible. As a further application, we obtain a combinatorial proof of Dunwoody's accessibility theorem of finitely presented groups., Comment: 9 pages

A digraph is connected-homogeneous if every isomorphism between two finite connected induced subdigraphs extends to an automorphism of the whole digraph. In this paper, we completely classify the countable connected-homogeneous digraphs., 49 pages

We classify the connected-homogeneous digraphs with more than one end. We further show that if their underlying undirected graph is not connected-homogeneous, they are highly-arc-transitive., 23 pages, 2 figures