# Howard Rowlee Lecture Series 1997

Efim Zelmanov, professor of Mathematics, Yale University, was awarded the Fields Medal (the mathematical analogue of the Nobel Prize) in 1994 for his seminal contributions to group theory.He will inaugurate the Howard Rowlee Lecture series with two lectures.

The Howard Rowlee Lecture series is made possible through a generous donation by Mr. Howard E. Rowlee, Jr., who has established a fund at the University of Nebraska Foundation to support research in mathematics. The inaugural lectures in this series are also supported by the College of Arts and Sciences Discrete and Experimental Mathematics program and the Department of Mathematics and Statistics.

## What makes a group infinite?

Abstract: Given a group presented by generators and relations, how can we say if it is finite or infinite? In 1964 Golod and Shafarevich suggested asufficient condition for such a group to be infinite. Zelmanov will discuss groups satisfying the Golod-Shafarevich condition.

April 29, 1997, Bessey Hall Auditorium, 4:00–5:00 p.m., Reception: 3:15–4:00 p.m., Bessey Hall 108

## On growth of algebras

Abstract: Zelmanov will discuss the notion of growth (Gelfand–Kirillov dimension) of algebras and some important examples of infinite dimensional algebrasof small growth.

April 30, 1997, Oldfather Hall 307, 4:00–5:00 p.m., UNL Refreshments: 3:15–4:00 p.m., Oldfather Hall 807

There will be a workshop in geometric group theory to be held on the campus of the University of Nebraska at Lincoln on April 28–30, 1997 in connection with the Howard Rowlee Lectures. We anticipate approximately six to ten speakers at the workshop. All who are interested are welcome to attend both the Howard Rowlee Lectures and the workshop.

Here are some of the main speakers which will participate in the Workshop, titles and abstracts of their talks.

### Tits-buildings, Gromov-boundary and the quasi-isometry problem for a certain class of Coxeter groups

Nadia Benakli, Columbia University (New York)

We will consider examples of Coxeter groups that are quasi-isometric to Tits-buildings. To classify these Coxeter groups up to quasi-isometry, we will use the combinatorial structure of the corresponding Tits-buildings and the topological properties of their Gromov-boundary when they are Gromov-hyperbolic.

### Infinite families of aspherical 2-complexes with the same Euler characteristic

Paul Brown, University of California-Berkeley

Let p be an odd prime. There is a countably infinite family of aspherical, compact 2-complexes X_i with word hyperbolic fundamental group and a finite group of automorphisms which is transitive on 2-cells such that the Euler characteristic of X_i is p. The construction uses a particular family of Cayley graphs for dihedral groups, embedded on surfaces, to construct polygons of finite groups.

### Non-uniform Lattices on Uniform Trees

Lisa Carbone, Columbia University (New York)

Let X be a locally finite tree. Then G=Aut(X) is a locally compact group. Following [BK] and [BL], a subgroup L of G is discrete if all vertex stabilizers L_x are finite. We then call L a lattice if the volume

Vol(L\\X)=\sum_{x\in L\X} 1/|L_x |

is finite, and a uniform lattice if, in addition, the quotient graph L\X is finite.

A non-uniform lattice L in G=Aut(X) is a discrete subgroup L of G acting on X with infinite quotient L\X and finite covolume Vol(L\\X).

We wish to understand when a locally finite tree admits a non-uniform lattice.

It is known ([BK]) that if G is unimodular, and G\X is finite, then G contains a uniform lattice. H. Bass and A. Lubotzky conjecture that under some additional assumptions, G also contains a non-uniform lattice.

Conjecture ([BL]):

Let X be a locally finite tree, and let G=Aut(X). Suppose that G is unimodular and G\X is finite. If G is not discrete and G acts minimally on X, then there is a non-uniform lattice L in G.

In this talk, we discuss a proof of this conjecture using the edge-indexed graph of a graph of groups for a tree-action.

We also discuss techniques for constructing tree-lattices; both uniform and non-uniform, on a locally finite tree X.

Bibliography:

[BK] Bass H and Kulkarni R, Uniform Tree Lattices, Journal of the Amer Math Society, vol 3 (4), 1990

[BL] Bass H and Lubotzky A Tree Lattices, to appear in Birkhauser, 1996

### TBA

Robert Jajcay, Indiana State University

A map on an orientable surface is said to be regular if its map automorphism group acts transitively on darts (i.e., edges with direction). A Cayley map is a Cayley graph embedded in an orientable surface such that the cyclic ordering, induced by a fixed surface orientation, of darts emanatingfrom each vertex yields the same cyclic permutation of generators and their inverses.

In our talk, we shall present several new constructions of infinite families of regular Cayley maps. In particular, applying the general theory of Cayley maps outlined in Richter and Siran (1996), we prove that each Cayley map is finitely covered by a regular Cayley map with the same distribution of inverses. Also, for any given degree and any distribution of inverses we characterize all face lengths for which there exists a regular Cayley map of this degree, face length, and distribution of inverses.

The presented material is a result of a joint work with Bruce Richter Jozef Siran.

### Algebraic Geometric Invariants of a class of Parafree Group

Sal Liriano, City College of CUNY

Given a finitely generated group G, the set Hom(G, SL_2 C) inherits the structure of an affine algebraic variety R(G) called the representation variety of G. The affine variety R(G) is an invariant of the finitely generated presentation of the group G. In fact, Hom(G,A) has the structure of an algebraic variety for any algebraic group A. The choice of SL_2 C is primarily since finitely generated free groups imbed into SL_2C, and every finitely generated group is a proper quotient of a finitely generated free group. If G is close'' to a free group, as in the case of a one-relator group, then one would expect R(G) to reflect the structure of G.

In this talk some invariants of R(G) will be introduced and a theorem giving the dimension of R(G) for a class of one-relator groups will be discussed and applied to a class of groups constructed by Gilbert Baumslag. Another invariant of R(G) will be discussed in relation to a class of groups containing as a subclass a class of knot groups. If time allows, the presentation will end with naive algebraic geometric proof of a celebrated theorem of W. Magnus.

### Bounded cohomology and metabolicity of negatively curved complexes.

Igor Mineyev, University of Utah:

The concept of a metabolic group was suggested by S.Gersten. A group is called metabolic if the second bounded cohomology H^2_{(\infty)}(X,A) vanishes for any normed abelian group A. There is a more revealing definition saying that G admits a combing with uniformly bounded areas of triangles. "Metabolic" implies "hyperbolic". One distinguishes Z-metabolicity and R-metabolicity. We are going to show that the fundamental group of a negatively curved complex is R-metabolic.