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of 177
pro vyhledávání: '"Elder, Murray"'
A connected undirected graph is called \emph{geodetic} if for every pair of vertices there is a unique shortest path connecting them. It has been conjectured that for finite groups, the only geodetic Cayley graphs which occur are odd cycles and compl
Externí odkaz:
http://arxiv.org/abs/2406.00261
A connected graph is called \emph{geodetic} if there is a unique geodesic between each pair of vertices. In this paper we prove that if a finitely generated group admits a Cayley graph which is geodetic, then the group must be virtually free. Before
Externí odkaz:
http://arxiv.org/abs/2311.03730
We call a graph $k$-geodetic, for some $k\geq 1$, if it is connected and between any two vertices there are at most $k$ geodesics. It is shown that any hyperbolic group with a $k$-geodetic Cayley graph is virtually-free. Furthermore, in such a group
Externí odkaz:
http://arxiv.org/abs/2211.13397
Testing isomorphism of infinite groups is a classical topic, but from the complexity theory viewpoint, few results are known. S{\'e}nizergues and the fifth author (ICALP2018) proved that the isomorphism problem for virtually free groups is decidable
Externí odkaz:
http://arxiv.org/abs/2110.00900
Autor:
Elder, Murray, Piggott, Adam
We show that groups presented by inverse-closed finite convergent length-reducing rewriting systems are characterised by a striking geometric property: their Cayley graphs are geodetic and side-lengths of non-degenerate triangles are uniformly bounde
Externí odkaz:
http://arxiv.org/abs/2106.03445
We extend work of the first author and Khoussainov to show that being Cayley automatic is closed under taking the restricted wreath product with a virtually infinite cyclic group. This adds to the list of known examples of Cayley automatic groups.
Externí odkaz:
http://arxiv.org/abs/2103.10648
Autor:
Elder, Murray, Piggott, Adam
We prove that a group is presented by finite convergent length-reducing rewriting systems where each rule has left-hand side of length 3 if and only if the group is plain. Our proof goes via a new result concerning properties of embedded circuits in
Externí odkaz:
http://arxiv.org/abs/2009.02885
We propose a new generalisation of Cayley automatic groups, varying the time complexity of computing multiplication, and language complexity of the normal form representatives. We first consider groups which have normal form language in the class $\m
Externí odkaz:
http://arxiv.org/abs/2008.02511
In contrast to being automatic, being Cayley automatic \emph{a priori} has no geometric consequences. Specifically, Cayley graphs of automatic groups enjoy a fellow traveler property. Here we study a distance function introduced by the first author a
Externí odkaz:
http://arxiv.org/abs/2008.02381
Autor:
Bishop, Alex, Elder, Murray
A direct consequence of Gromov's theorem is that if a group has polynomial geodesic growth with respect to some finite generating set then it is virtually nilpotent. However, until now the only examples known were virtually abelian. In this note we f
Externí odkaz:
http://arxiv.org/abs/2007.06834