CAPITAL REGION ALGEBRA/NUMBER THEORY SEMINAR 


Speaker:       Professor Jennifer Taback, Univ. at Albany
Time & Place:  Wednesday, November 29, at 4:30pm, in ES 152A  
Title:         An Introduction to Automatic Groups 


Abstract: 

The theory of automatic groups arose out of a desire to "see" the Cayley
graph of a finitely generated group.  By this we mean to have a picture of
the Cayley graph in a finite ball around the origin and some finite
machine which tells us how to piece togther copies of this ball to create
the entire graph.  Thurston formalized this idea using simple computing
machines called finite state automata to define automatic groups,
whose Cayley graphs can be constructed in this manner.  Automatic groups
have many nice algebraic and geometric properties, such as solvable word
problem, finite presentation and quadratic isoperimetric inequality.

I will define automatic groups using finite state automata as well as give
an alternate, more geometric definition.  Several examples will be
discussed, as well as a broadening of the definition to include
asynchronous automatic and combable groups.  It is often quite difficult
to decide if a class of groups is (asynchronously) automatic.  I will
outline the proof that hyperbolic groups are automatic.  

Lastly, I will discuss a class of groups of great interest to me,
$PSL_2 ( Z[1/p_1, ... , 1/p_r] )$, where the $p_i$ are prime.  
These groups have a distinguished "cusp" subgroup $\Gamma_r$.  
When $r=1$, the group $\Gamma_1$ is the solvable Baumslag-Solitar
group, which is asynchronously automatic.  
I will show that the $\Gamma_r$ are also asynchronously automatic.  
This will ivolve contsructing an interesting geometric model 
for these groups.



Refreshments at 4:15pm in ES 152.