Andrei Malyutin:
Pseudo-characters of braid groups
ABSTRACT:
This talk will be a brief exposition of results and problems related to
pseudo-characters of braid groups. We will present several examples of
braid group pseudo-characters and discuss their properties. The structure
of the space of braid group pseudo-characters will be described. We will
also explain the ways to implement braid group pseudo-characters in knot
theory and in low-dimensional dynamical systems.
A functional $f: G \to \R$ on a group $G$ is called a {\it
pseudo-character\/} (or {\it pseudo-homomorphism\/}, or {\it homogeneous
quasimorphism\/}) if the set $\{f(ab)-f(a)-f(b)\ :\ a,b\in G\}$ is bounded
and, moreover, we have $f(a^k)=k f(a)$ for each integer $k$ and each $a\in
G$. It follows immediately from this definition that the pseudo-characters
of a group constitute a linear vector space. Results of M.Bestvina and
K.Fujiwara imply that the space of pseudo-characters of the braid group
$B_n$ is infinite-dimensional if $n>2$. Nevertheless, looking-for a
specific braid group pseudocharacter and an algorithm for its computation
is a hard and interesting problem. Only a few series of braid group
pseudo-characters are at present described explicitly. One of these series
can be defined through Thurston's hyperbolic construction and the Poincare
translation number or, more briefly, via the Dehornoy ordering. Another
one is close to the signature of knots and links and appears in works of
J.M.Gambaudo and E.Ghys. Ghys suggested using braid group
pseudo-characters to construct new invariants of low-dimensional dynamical
systems. Pseudocharacters are also useful for studying random walks on the
braid group and random links. It turns out that each braid group
pseudo-character provides certain conditions for admissibility of the
Markov destabilization and other braid moves. Combining this fact with
results of Birman, Menasko, and Dynnikov allows us to implement
pseudocharacters for recognition of prime knots and links.