The Conjugacy Problem for Graph Products with Cyclic Edge Groups

A group G is S-universal if every countable group can be embedded in a quotient of G.In this paper we give conditions under which HNN groups are S-universal.These are then applied to showing the S-universality of certain classes of Kleinian groups (discontinuous subgroups of PSL^C)).Finally some connections are mentioned between Sg-universality and the theory of ends.Introduction.A group G is So-universal if every countable group can be embedded in a quotient of G. Thus all nonabelian free groups are Suniversal [4], while it is known that all nontrivial free products except the infinite dihedral group [10], and large classes of amalgamated free products are also , [11].Sacerdote and Schupp have extended these to certain HNN and one-relator groups [9].In this paper we give a series of results which complement those of Sacerdote and Schupp but are somewhat less technical and more directly applicable to geometrically and arithmetically defined groups [2], [3].This will be applied in 3 to showing the S?-universality of a class of Kleinian groups.Finally we make some observations on the relationship between SG;-universality and the theory of ends.1. Recall that an HNN group is a group of the form G = </" ...,t",K; rel K, t^tr1 = <P,(A)>where {L,} are subgroups of K, and {<p,} are isomorphisms L, - <p,(jL,) < K. K is called the base, {L, <p(L)} associated subgroups and </" . .., i") the free part.Groups of this type were used by Higman, Neumann and Neumann to show the S-universality of free groups of rank 2 [4].The base K can be embedded isomorphically in G, and a subgroup structure theory for HNN groups, paralleling that for amalgamated free products, has been developedby Karrass and Solitar [5].Now G has a free quotient of rank n if the free part has rank n, and since clearly a group with an S-universal quotient is itself S-universal we getLemma.An HNN group with free part of rank > 1 is SQ-universal.

We prove that if $G$ is a group that splits as a finite graph of finitely generated free groups with cyclic edge groups, and $G$ has no non-Euclidean Baumslag-Solitar subgroups, then $G$ is the fundamental group of a compact nonpositively curved cube complex. In addition, if $G$ is also word-hyperbolic (i.e., if $G$ contains no Baumslag-Solitar subgroups of any type), we show that $G$ is linear (in fact, is a subgroup of $SL_n(Z)$).

An infinite graph Γ is minor excluded if there is a finite graph that is not a minor of Γ. We prove that minor excluded graphs have finite Assouad–Nagata dimension and study minor exclusion for Cayley graphs of finitely generated groups. Our main results and observations are: (1) minor exclusion is not a group property: it depends on the choice of generating set; (2) a group with one end has a generating set for which the Cayley graph is not minor excluded; (3) there are groups that are not minor excluded for any set of generators, like ℤ3; (4) minor exclusion is preserved under free products; and (5) virtually free groups are minor excluded for any choice of finite generating set.

On the Metric Dimension of Infinite Graphs

Optimal identifying codes of two families of Cayley graphs

Generalising some results about right-angled Artin groups to graph products of groups

Amenability of Schreier graphs and strongly generic algorithms for the conjugacy problem

Decision problems in classes of finitely presented groups with solvable conjugacy problem, with solvable order problem, and with solvable power problem are considered. It is shown that the problem of recognizing group properties is in no real sense improved by restricting attention to these classes.

For a finite presentation of a group, or more generally, a two-complex, we define a function analogous to the Dehn function that we call the annular Dehn function. This function measures the combinatorial area of maps of annuli into the complex as a function of the lengths of the boundary curves. A finitely presented group has solvable conjugacy problem if and only if its annular Dehn function is recursive.As with standard Dehn functions, the annular Dehn function may change with change of presentation. We prove that the type of function obtained is preserved by change of presentation. Further we obtain upper bounds for the annular Dehn functions of free products and, more generally, amalgamations or HNN extensions over finite subgroups.

&lt;p style="text-align: left;" dir="ltr"&gt; Let &lt;em&gt;G&lt;/em&gt; be a finite group and let &lt;em&gt;N&lt;/em&gt; be a fixed normal subgroup of &lt;em&gt;G&lt;/em&gt;. In this paper, a new kind of graph on &lt;em&gt;G&lt;/em&gt;, namely the intersection graph is defined and studied. We use &lt;img src="/public/site/images/ikhsan/equation.png" alt="" width="6" height="4" /&gt; to denote this graph, with its vertices are all normal subgroups of &lt;em&gt;G&lt;/em&gt; and two distinct vertices are adjacent if their intersection in &lt;em&gt;N&lt;/em&gt;. We show some properties of this graph. For instance, the intersection graph is a simple connected with diameter at most two. Furthermore we give the graph structure of &lt;img src="/public/site/images/ikhsan/equation_(1).png" alt="" width="6" height="4" /&gt; for some finite groups such as the symmetric, dihedral, special linear group, quaternion and cyclic groups. &lt;/p&gt;

Counting 1-factors in infinite graphs

We use the theory of group actions on profinite trees to prove that the fundamental group of a finite, 1-acylindrical graph of free groups with finitely generated edge groups is conjugacy separable. This has several applications: we prove that positive, C′(1/6) one-relator groups are conjugacy separable; we provide a conjugacy separable version of the Rips construction; we use this latter to provide an example of two finitely presented, residually finite groups that have isomorphic profinite completions, such that one is conjugacy separable and the other does not even have solvable conjugacy problem.

By the method of nonstandard analysis, the definition of *-finite graph is given, and necessary and sufficient conditions of *-finite graph are obtained. Further, by the Transfer Principle, we apply the theory of finite graph to *-finite graph, embed given infinite graph into some *-finite graph, and finally obtain the related results of infinite graph.

We present a metric condition $\TTMetric$ which describes the geometry of classical small cancellation groups and applies also to other known classes of groups such as two-dimensional Artin groups. We prove that presentations satisfying condition $\TTMetric$ are diagrammatically reducible in the sense of Sieradski and Gersten. In particular, we deduce that the standard presentation of an Artin group is aspherical if and only if it is diagrammatically reducible. We show that, under some extra hypotheses, $\TTMetric$-groups have quadratic Dehn functions and solvable conjugacy problem. In the spirit of Greendlinger's lemma, we prove that if a presentation P = 〈X| R〉 of group G satisfies conditions $\TTMetric -C'(\frac {1}{2})$, the length of any nontrivial word in the free group generated by X representing the trivial element in G is at least that of the shortest relator. We also introduce a strict metric condition $\TTMetricStrict$, which implies hyperbolicity.

Cycles through all finite vertex sets in infinite graphs