## Topology & Geometric Group Theory Seminar

## Fall 2007

### 1:30 - 2:30, Malott 253

Tuesday, September 4

**Roland
Roeder**, University of Toronto

*Computing arithmetic invariants for hyperbolic reflection groups*

The following is collaborative work with Omar Antolin-Camarena and
Gregory Maloney from the University of Toronto.

I will demonstrate a collection of computer scripts written in PARI/GP
to compute, for reflection groups determined by finite-volume
polyhedra in **H**^{3}, the commensurability invariants
known as the invariant trace field and invariant quaternion algebra.
These scripts also allow one to determine arithmeticity of such groups
and the isomorphism class of the invariant quaternion algebra by
analyzing its ramification.

I present many computed examples of these invariants. This is enough
to show that most of the groups that we consider are pairwise
incommensurable. For pairs of groups with identical invariants, not
all is lost: when both groups are arithmetic, having identical
invariants guarantees commensurability. We discover many "unexpected"
commensurable pairs this way. We also present a non-arithmetic pair
with identical invariants for which we cannot determine
commensurability.

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