319 research outputs found
Extremal Infinite Graph Theory
We survey various aspects of infinite extremal graph theory and prove several
new results. The lead role play the parameters connectivity and degree. This
includes the end degree. Many open problems are suggested.Comment: 41 pages, 16 figure
Splitting Line Patterns in Free Groups
We construct a boundary of a finite rank free group relative to a finite list
of conjugacy classes of maximal cyclic subgroups. From the cut points and
uncrossed cut pairs of this boundary we construct a simplicial tree on which
the group acts cocompactly. We show that the quotient graph of groups is the
JSJ decomposition of the group relative to the given collection of conjugacy
classes.
This provides a characterization of virtually geometric multiwords: they are
the multiwords that are built from geometric pieces. In particular, a multiword
is virtually geometric if and only if the relative boundary is planar.Comment: 22 pages, 6 figures; v2 fixed a few typos; v3 38 pages, 21 figures;
v4 30 pages, 11 figures 'Preliminaries' section expanded to make paper
self-contained and split into two sections. Some arguments refactored and
simplified. Paper streamlined; v5 56 pages, 21 figures Added examples and
improved exposition according to referee comments. To appear in Algebraic &
Geometric Topolog
Combinatorial Properties of Finite Models
We study countable embedding-universal and homomorphism-universal structures
and unify results related to both of these notions. We show that many universal
and ultrahomogeneous structures allow a concise description (called here a
finite presentation). Extending classical work of Rado (for the random graph),
we find a finite presentation for each of the following classes: homogeneous
undirected graphs, homogeneous tournaments and homogeneous partially ordered
sets. We also give a finite presentation of the rational Urysohn metric space
and some homogeneous directed graphs.
We survey well known structures that are finitely presented. We focus on
structures endowed with natural partial orders and prove their universality.
These partial orders include partial orders on sets of words, partial orders
formed by geometric objects, grammars, polynomials and homomorphism orders for
various combinatorial objects.
We give a new combinatorial proof of the existence of embedding-universal
objects for homomorphism-defined classes of structures. This relates countable
embedding-universal structures to homomorphism dualities (finite
homomorphism-universal structures) and Urysohn metric spaces. Our explicit
construction also allows us to show several properties of these structures.Comment: PhD thesis, unofficial version (missing apple font
When is a polynomially growing automorphism of geometric ?
The main result of this paper is an algorithmic answer to the question raised
in the title, up to replacing the given by a positive
power.
In order to provide this algorithm, it is shown that every polynomially
growing automorphism can be represented by an iterated Dehn twist
on some graph-of-groups with . One then uses
results of two previous papers \cite{KY01, KY02} as well as some classical
results such as the Whitehead algorithm to prove the claim
LIPIcs, Volume 248, ISAAC 2022, Complete Volume
LIPIcs, Volume 248, ISAAC 2022, Complete Volum
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