# Complexity of 3-manifolds and groups

*by Professor Carlo Petronio*

*Institution:*University of Pisa

*Date: Mon 20th November 2006*

*Time: 3:15 PM*

*Location: Room 213 Richard Berry Building*

*Abstract*: The complexity c(M) of a closed 3-manifold M is an integer that measures how complicated M is. Matveev has shown that for any given n only finitely many prime M's have c(M)=n. However the exact value of c(M) has been computed only for a (huge but) finite number of prime M's (namely, those with c(M) at most 12). In this seminar I will talk about connections between c(M) and vol(M) for a hyperbolic M, and I will exhibit three infinite families of such M's with explicit asymptotic two-sided bounds on c(M), where the upper and lower bounds differ by a fixed linear map.

I will also discuss the notion of complexity c(G) for a finitely presented group G, and the related invariant T(G) introduced by Delzant. c(G) measures the minimal total length of the relations in a presentation of G, and T(G) is defined similarly using triangular presentations. I will mention some connections between c(M), c(G) and T(G) when G is the fundamental group of M, and I will give various estimates on the values of c(G) and T(G) when G is Abelian and when G is a Milnor group (i.e. a finite one acting orthogonally on the 3-sphere). Some of these estimates depend on other ones that I will give for some very special types of elliptic 3-manifolds.

This is a report of joint work with Matveev-Vesnin, Vesnin, and Pervova.

*For More Information:* Dr. Lawrence Reeves email: L.Reeves@ms.unimelb.edu.au