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Matthias Hamann

Publications and source records attributed to Matthias Hamann.

At least 37 records · Page 2Linked to original sources
arXiv2019

The planar Cayley graphs are effectively enumerable I: consistently planar graphs

We obtain an effective enumeration of the family of finitely generated groups admitting a faithful, properly discontinuous action on some 2-manifold contained in the sphere. This is achieved by introducing a type of group presentation capturing exactly these groups. Extending this in a companion paper, we find group presentations capturing the planar finitely generated Cayley graphs. Thus we obtain an effective enumeration of these Cayley graphs, yielding in particular an affirmative answer to a question of Droms et al.

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arXiv2019

The planar Cayley graphs are effectively enumerable II

We show that a group admits a planar, finitely generated Cayley graph if and only if it admits a special kind of group presentation we introduce, called a planar presentation. Planar presentations can be recognised algorithmically. As a consequence, we obtain an effective enumeration of the planar Cayley graphs, yielding in particular an affirmative answer to a question of Droms et al. asking whether the planar groups can be effectively enumerated.

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arXiv2018

Tree amalgamations and quasi-isometries

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.

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arXiv2017

Self-embeddings of trees

We prove a fix point theorem for monoids of self-embeddings of trees. As a corollary, we obtain a result by Laflamme, Pouzet and Sauer that a tree either contains a subdivided binary tree as a subtree or has a vertex, and edge, an end or two ends fixed by all its self-embeddings.

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arXiv2016

Planar transitive graphs

We prove that the first homology group of every planar locally transitive finite graph $G$ is a finitely generated ${\rm Aut}(G)$-module and we prove a similar result for the fundamental group of locally finite planar Cayley graphs. Corollaries of these results include Droms's theorem that planar groups are finitely presented and Dunwoody's theorem that planar locally finite transitive graphs are accessible.

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arXiv2015

$k$-Blocks: a connectivity invariant for graphs

A $k$-block in a graph $G$ is a maximal set of at least $k$ vertices no two of which can be separated in $G$ by fewer than $k$ other vertices. The block number $β(G)$ of $G$ is the largest integer $k$ such that $G$ has a $k$-block. We investigate how $β$ interacts with density invariants of graphs, such as their minimum or average degree. We further present algorithms that decide whether a graph has a $k$-block, or which find all its $k$-blocks. The connectivity invariant $β(G)$ has a dual width invariant, the block-width ${\rm bw}(G)$ of $G$. Our algorithms imply the duality theorem $β= {\rm bw}$: a graph has a block-decomposition of width and adhesion $< k$ if and only if it contains no $k$-block.

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arXiv2015

Bounding connected tree-width

Diestel and Müller showed that the connected tree-width of a graph $G$, i.e., the minimum width of any tree-decomposition with connected parts, can be bounded in terms of the tree-width of $G$ and the largest length of a geodesic cycle in $G$. We improve their bound to one that is of correct order of magnitude. Finally, we construct a graph whose connected tree-width exceeds the connected order of any of its brambles. This disproves a conjecture by Diestel and Müller asserting an analogue of tree-width duality.

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arXiv2014

Generating the cycle space of planar graphs

We prove that the cycle space of every planar finitely separable 3-connected graph $G$ is generated by some ${\rm Aut}(G)$-invariant nested set of cycles. We also discuss the situation in the case of smaller connectivity.

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arXiv2014

Accessibility in transitive graphs

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.

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arXiv2014

Canonical tree-decompositions of finite graphs I. Existence and algorithms

We construct tree-decompositions of graphs that distinguish all their k-blocks and tangles of order k, for any fixed integer k. We describe a family of algorithms to construct such decompositions, seeking to maximize their diversity subject to the requirement that they commute with graph isomorphisms. In particular, all the decompositions constructed are invariant under the automorphisms of the graph.

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arXiv2014

Canonical tree-decompositions of finite graphs II. Essential parts

In Part I of this series we described three algorithms that construct canonical tree-decompositions of graphs which distinguish all their k-blocks and tangles of order k. We now establish bounds on the number of parts in these decompositions that contain no such block or tangle, and determine conditions under which such parts contain nothing but a k-block.

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arXiv2013

Countable connected-homogeneous digraphs

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.

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arXiv2013

Homogeneous 2-partite digraphs

We call a 2-partite digraph D homogeneous if every isomorphism between finite induced subdigraphs that respects the 2-partition of D extends to an automorphism of D that does the same. In this note, we classify the homogeneous 2-partite digraphs.

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arXiv2013

Spanning trees in hyperbolic graphs

In this paper we construct spanning trees in hyperbolic graphs that represent their hyperbolic compactification in a good way: so that the tree has a bounded number of distinct rays to each boundary point. The bound depends only on the (Assouad) dimension of the boundary. As a corollary we sharpen a result of Gromov which says that from every hyperbolic graph with bounded degrees one can construct a tree outside the graph with a continuous surjection from the ends of the tree onto the hyperbolic boundary such that the surjection is finite-to-one. We will construct a tree with these properties inside the hyperbolic graph, which in addition is also a spanning tree of that graph.

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arXiv2013

Group actions on metric spaces: fixed points and free subgroups

We look at group actions on metric spaces, particularly at group actions on geodesic hyperbolic spaces. We classify the types of automorphisms on these spaces and prove several results about the density of the hyperbolic limit set of the group in the whole limit set of the group. In the case of graphs, our theorems hold also when the graphs are not hyperbolic.

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arXiv2011

On fixing boundary points of transitive hyperbolic graphs

We show that there is no 1-ended, planar, hyperbolic graph such that the stabilizer of one of its hyperbolic boundary points acts transitively on the vertices of the graph. This gives a partial answer to a question by Kaimanovich and Woess.

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arXiv2011

On the tree-likeness of hyperbolic spaces

In proper hyperbolic geodetic spaces we construct rooted $\mathbb R$-trees with the following properties. On the one hand, every ray starting at the root is quasi-geodetic; so these $\mathbb R$-trees represent the space itself well. At the same time, the trees boundary reflects the boundary of the space in that the number of disjoint rays to a boundary point is bounded in terms of the (Assouad) dimension of the hyperbolic boundary.

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