Symmetry breaking involves coloring the elements of a structure so that the only automorphism which respects the coloring is the identity. We investigate how much information we would need to be able to compute a 2-coloring of a computable finite-branching tree under the predecessor function which eliminates all automorphisms except the trivial one; we also generalize to n-colorings for fixed n...

We show that all the tangles in a finite graph or matroid can be distinguished by a single tree-decomposition that is invariant under the automorphisms of the graph or matroid. This comes as a corollary of a similar decomposition theorem for more general combinatorial structures, which has further applications. These include a new approach to cluster analysis and image segmentation. As another ...

Automorphisms of handlebodies arise naturally in the classification of automorphisms of three-manifolds. Among automorphisms of handlebodies, there are certain automorphisms called irreducible (or generic), which are analogues of pseudo-Anosov automorphisms of surfaces. We show that irreducible automorphisms of handlebodies exist and develop methods for constructing a range of examples.

this article is intended to be a survey on some combinatorial topics in group theory. the bibliography at the end is neither claimed to be exhaustive, nor is it necessarily connected with a reference in the text. i include it as i see it revolves around the concepts which are discussed in the text.

Let G be a finite p-group and let Aut_l(G) be the group of absolute central automorphisms of G. In this paper we give necessary and sufficient condition on G such that each absolute central automorphism of G fixes the centre element-wise. Also we classify all groups of orders p^3 and p^4 whose absolute central automorphisms fix the centre element-wise.

Panagiotou and Stufler (arXiv:1502.07180v2) recently proved one important fact on their way to establish the scaling limits of random Pólya trees: a uniform random Pólya tree of size n consists of a conditioned critical Galton-Watson tree Tn and many small forests, where with probability tending to one as n tends to infinity, each forest Fn(v) is maximally of size |Fn(v)| = O(log n). Their proo...

We study the automorphism groups of free-by-cyclic and show these are finitely generated in following cases: (i) when defining has linear growth (ii) rank underlying free group at most 3. The techniques we use actions on trees, including trees cylinders due to Guirardel Levitt, relative hyperbolicity (due Gautero Lustig, Ghosh, Dahmani Li) filtration automorphisms a preserving tree, Bass Jiang,...

To most mathematicians and computer scientists the word "tree" conjures up, in addition to the usual image, the image of a connected graph with no circuits. We shall deal with various aspects and generalizations of these mathematical trees. (As Peter Shalen has pointed out, there will be leaves and foliations in this discussion, but they do not belong to the trees!) In the last few years variou...

To most mathematicians and computer scientists the word " tree " conjures up, in addition to the usual image, the image of a connected graph with no circuits. We shall deal with various aspects and generalizations of these mathematical trees. (As Peter Shalen has pointed out, there will be leaves and foliations in this discussion, but they do not belong to the trees!) In the last few years vari...