[Binghamton Geometry/Topology Seminar] October 8

[Marco Varisco]

Binghamton Geometry/Topology Seminar
http://www.math.binghamton.edu/dept/topsem/

Date: Thursday, October 8, 2009

Time: 2:50 pm

Place: Library North 2205
(followed by coffee/tea in the Anderson Reading Room)

Speaker: Dan Farley (Miami University, Ohio)

Title: Finiteness properties of groups acting on compact ultrametric spaces

Abstract: (Joint with Bruce Hughes.) Let T be a locally finite rooted
simplicial tree. The space of ends of T, which we denote X, is a
compact ultrametric space. A finite similarity structure S(X) assigns
a finite (possibly empty) set S(B1,B2) of surjective similarities j:
B1 → B2 to each pair of balls B1, B2 ⊂ X. The union of all sets
S(B1,B2) (for varying B1 and B2) is also assumed to satisfy certain
groupoid-like properties.
We are interested in the groups determined by finite similarity
structures. Given S(X) (as above), we let G(S) denote the group of all
self-homeomorphisms of X that are locally determined by S(X). That is,
if h ∈ G(S), then, for every x ∈ X, there are balls B1 and B2, and a j
∈ S(B1,B2), such that x ∈ B1 and j|B1=h|B1.
For example, if we let T be the ordered infinite rooted binary tree, X
be its space of ends, and, for any balls B1 and B2, S(B1,B2) be the
singleton set containing the unique order-preserving similarity from
B1 to B2, then G(S) is Thompson's group V. It appears that there are
many other examples as well.
I will discuss joint work in progress with Bruce Hughes, in which we
argue that a fairly general class of the groups G(S) have type F∞,
i.e., have a classifying complex with finitely many cells in each
dimension.