(Reposted from my old ldt blog)
Here’s an interesting paper, which appeared on the ArXiv a couple of weeks ago: “The Casson invariant and the word metric on the Torelli group” by Broaddus, Farb and Putman . This paper continues a line of inquiry started by Morita  in 1991. Given a Heegaard splitting for a homology 3-sphere, we can think of the manifold as being the result of cutting the 3-sphere open along a Heegaard surface of the same genus, then re-gluing along a new map. Morita showed that you can get any Heegaard splitting for any homology sphere by composing the original gluing map for the 3-sphere with an element of the Torelli group. (A surface homomorphism is in the Torelli group if the induced automorphism of the first homology group of the surface is the identity.) Morita then showed that one can compute the Casson invariant from this regluing map.
Broaddus, Farb and Putman have shown that for any fixed generating set for the Torelli group, the Casson invariant of a manifold determined by a given re-gluing map is bounded above by a constant times the square of the distance from the origin in the word metric. (The word metric for a group is the path metric for a Cayley graph defined by the generating set.) They also demonstrate that there is sequence of elements of the Torelli group determining manifolds whose Casson invariants grow asymptotically quadratically with the distance from the origin. Thus a quadratic bound is the best that one can hope for.
Consider the map from the Torelli group to the (non-negative) integers that sends each element to the Casson invariant of the homology 3-sphere it defines. Broaddus, Farb and Putman’s result implies that as you follow a path through a Cayley graph for the Torelli group, the numbers at each element grow at most quadratically. However, there are many paths for which the numbers will not grow at all. For example, if the re-gluing map extends to an automorphism of either handlebody then the resulting manifold is the 3-sphere, so the Casson invariant is zero. There are also many homology spheres for which the Casson invariant is zero (see for example ). It would be interesting to classify the set of all automorphism that map to zero. This along with a classification of the Torelli automorphisms that extend into a given handlebody (I don’t know if this is known.) might lead to a nice characterization of the set of homology 3-spheres with trivial Casson invariants.
 and , follow links
 S. Morita, On the structure of the Torelli group and the Casson invariant, Topology 30 (1991), no. 4, 603–621.