Abstracts

Group Theory, Actions and Computation

Nonabelian tensor squares of free nilpotent groups of finite rank
by
Russell Blyth
Saint Louis University
Coauthors: Primož Moravec (Univerza v Ljubljani), Robert Fitzgerald Morse (University of Evansville)

Let G be any group. Then the group G⊗G generated by the symbols g⊗h, where g, h ∈ G, subject to the relations

gh⊗k = (gh⊗gk)(g⊗k)     and     g⊗hk = (g⊗h)(hg⊗hk)

for all g, h, and k in G, where ^xy = xyx^-1 for x, y ∈ G, is called the nonabelian tensor square of G. Let ∇(G) be the central subgroup of G⊗G generated by the set {g⊗g | g ∈ G}. The factor group G⊗G/∇(G) is called the nonabelian exterior square of G, denoted by G∧G. We discuss results concerning the structures of the nonabelian tensor squares and the nonabelian exterior squares of the free nilpotent groups of finite rank. These results were motivated and guided by computations performed using GAP.

Date received: October 19, 2007

Partially symmetric automorphisms of free groups
by
Ruth Charney
Brandeis University
Coauthors: Kai-Uwe Bux, Adam Piggott, and Karen Vogtmann

We compute the virtual cohomological dimension (vcd) of the group of outer automorphisms of a free group which fix certain generators up to conjugacy. The technique is to find a retraction of Culler--Vogtmann's ``outer space" preserved by this group. As a corollary, we compute the vcd of the outer automorphism group of right-angled Artin groups associated to trees.

Date received: November 2, 2007

Short presentations for the alternating and symmetric groups
by
Marston Conder
University of Auckland
Coauthors: John Bray (Queen Mary, London), Charles Leedham-Green (Queen Mary, London), Eamonn O'Brien (U Auckland)

A standard presentation for the symmetric group S_n is given in terms of transpositions t_i = (i, i+1) for 1 ≤ i < n and the Coxeter relations satisfied by these. The number of generators is linear in n, and the number of relations is quadratic in n. I will describe some new presentations for S_n that involve a fixed number of generators and relations, and how these can be used to obtain short presentations for both the alternating groups A_n and the symmetric groups S_n, and then similarly for the finite classical linear groups.

Date received: November 20, 2007

Bicontactual rotary hypermaps
by
Antonio Breda d'Azevedo
University of Aveiro
Coauthors: Ilda Rodrigues

We present the classification of the orientably regular hypermaps that are bicontactual, that is, each face has only two adjacent faces. The classification of bicontactual regular maps (orientable and not orientable) was done by Wilson in 1976.

Date received: May 7, 2007

On Some Finiteness Properties in Infinite Groups
by
Benjamin Fine
Fairfield University, Fairfield, CT. 06840 U.S.A.
Coauthors: Gilbert Baumslag,Oleg Bogopolski,Anthony Gaglione,Gerhard Rosenberger,Dennis Spellman

We consider some questions concerning certain finiteness properties in infinite groups which are related to Marshall Hall's Theorem. We call these properties Property S and Property R and both are trivially true in finite groups. We give several elementary proofs using these properties for results on finitely generated subgroups of free groups and in limit groups as well as a new elementary proof of Marshall Hall's basic result. We next consider these properties within surface groups and prove an analog of Marshall Hall's theorem in that context. Finally we show that nilpotent groups and certain finite extensions of nilpotent groups satisfy these properties.

Date received: October 28, 2007

The strong symmetric genus and generalized symmetric groups: results and a conjecuture
by
Michael A. Jackson
Grove City College, USA

The strong symmetric genus of a finite group G is the smallest genus of a closed orientable topological surface on which G acts faithfully as a group of orientation preserving automorphisms. Marston Conder found the strong symmetric genus of the alternating and symmetric groups. The idea of the symmetric groups, S_n, can be expanded to the generalized symmetric groups, which are defined as G(n, m) is the wreath product of Z_m by S_n, where n, m ∈ Z₊. This puts the standard symmetric groups as a family of generalized symmetric groups, i.e. S_n = G(n, 1). Recently, the author has found the strong symmetric genus of the hyperoctahedral groups (which are the generalized symmetric groups of type G(n, 2)) and the groups of type G(n, 3). This talk will discuss these results as well as some additional cases of the strong symmetric genus of G(n, m) for m > 3. In addition a conjecture concerning the general results will be discussed.

Date received: October 19, 2007

Total chirality of maps and hypermaps on Riemann surfaces
by
Gareth Jones
University of Southampton
Coauthors: Antonio Breda

By Belyi's Theorem, the compact Riemann surfaces defined over algebraic number fields are those uniformised by subgroups of triangle groups, or equivalently obtained from hypermaps. The most symmetric of these correspond to normal subgroups of triangle groups, or equivalently to orientably regular hypermaps. Such a hypermap is termed chiral if it is not isomorphic to its mirror image. The most extreme form of this phenomenon is total chirality, where the hypermap and mirror image have no nontrivial common quotients.

Antonio Breda (Aveiro) and I have classified the totally chiral hypermaps of genus up to 1001. The least genus of any totally chiral hypermap is 211, attained by twelve orientably regular hypermaps with automorphism group A₇ and type (3, 4, 4) (up to triality). The least genus of any totally chiral map is 631, attained by a chiral pair of orientably regular maps of type {11, 4}, together with their duals; their automorphism group is the Mathieu group M₁₁. This is also the least genus of any totally chiral hypermap with non-simple automorphism group, in this case the perfect triple covering 3.A₇ of A₇.

Date received: November 1, 2007

Enumerating chiral maps on surfaces with a given underlying graph
by
Jin Ho Kwak
Combinatorial and Computational Mathematics Center, Pohang University of Science and Technology, Pohang, 790-784 Korea
Coauthors: Yan-Quan Feng and Jin-Xin Zhou

Abstract

Date received: October 28, 2007

Capable groups of class two and prime exponent
by
Arturo Magidin
University of Louisiana - Lafayette

A group G is capable if and only if G ≅ H/Z(H) for some group H. If G is a group of class two and prime exponent, capability can be characterised in terms of a closure operator on the lattice of subspaces of certain finite dimensional vector space over a field of p elements. I have been working towards a characterisation of the capable groups in this class via this equivalence. In the case of 5-generated groups, GAP was used to search through examples of non-closed subspaces; by considering these examples and why they were not closed, I was able to prove that the only non-capable groups among the 5-generated groups of class at most two and exponent p are the cyclic group and the groups that can be expressed as a direct product of two nonabelian groups G₁ and G₂ amalgamated over a subgroup of order p of the commutator subgroups. I will discuss these and other results, as well as the role GAP is playing in the investigations.

Date received: July 24, 2007

A census of edge-transitive tessellations
by
Tomaž Pisanski
IMFM, University of Ljubljana, Slovenia

B. Grünbaum and G. C. Shephard have classified edge-transitive tessellations according to their edge-symbol <p, q;k, l>. The growth rate of Bilinski diagrams for each of these tessellations has been determined by S. Graves, T.Pisanski and M.E. Watkins recently. We compute the number of edge-transitive tessellations for a given growth rate and present a census of these tessellations.

Date received: October 29, 2007

The Tits alternative for sherical generalized tetrahedron groups
by
Gerhard Rosenberger
Fachbereich Mathematik, University of Dortmund
Coauthors: B. Fine, V. gr.Rebel and H. Hulpke

A generalized tetrahedron group is defined to be a group G admitting the following presentation:

< x, y, z : xl = ym = zn = W1p(x, y) = W2q(y, z) = W3r(x.z) = 1 > .

These groups appear in many contexts, not least as fundamental groups of certain hyperbolic orbifolds or as subgroups of generalized triangle groups. If (p, q, r) is not (2, 2, 2) then the Tits alternative holds for G, that is, G contains a non-abelian free subgroup or is solvable-by-finite. If (p, q, r) = (2, 2, 2) we have many partial results, especially we give the the list of the finite generalized tetrahedron groups.

Date received: April 4, 2007

Old and new on the universal covering group of SL(2, R).
by
Gunter Steinke
Department of Mathematics and Statistics, University of Canterbury, Christchurch
Coauthors: Rainer Loewen

The structure and properties of the universal covering group [(W)\tilde] of SL(2, R) are well understood. However, since this group permits no faithful linear representation, it remains elusive and only a few geometries are known on which [(W)\tilde] acts as a group of automorphisms. We survey some known results and present a new geometry which essentially is determined by the one-parameter subgroups of [(W)\tilde] extended by a factor R.

Date received: September 6, 2007