##### Department of Mathematics,

University of California San Diego

****************************

### Math 295 - Mathematics Colloquium

## Lizhen Ji

#### University of Michigan

## Coarse Schottky Problem and Equivariant Cell Decomposition of Teichmuller Space

##### Abstract:

In this talk, I will explain some similar results and interaction between locally symmetric spaces and moduli spaces of Riemann surfaces. For example, let $M_g$ be the moduli space of Riemann of genus $g$, and $A_g$ be the moduli space of principally polarized abelian varieties of dimension $g$, i.e., the quotient of the Siegel upper space by $Sp(2g, Z)$. Then there is a Jacobian map $J: M_g \to A_g$, by associating to each Riemann surface its Jacobian. The celebrated Schottky problem is to characterize the image $J(M_g).$ Buser and Sarnak viewed $A_g$ as a complete metric space and showed that $J(M_g)$ lies in a very small neighborhood of the boundary of $A_g$ as $g$ goes to infinity. Motivated by this, Farb formulated the coarse Schottky problem: determine the image of $J(M_g)$ in the asymptotic cone (or tangent space at infinity) $C_\infty(A_g)$ of $A_g$, as defined by Gromov in large scale geometry. In a joint work with Enrico Leuzinger, we showed that $J(M_g)$ is $c$-dense in $A_g$ for some constant $c=c(g)$ and hence its image in the asymptotic cone $C_\infty(A_g)$ is equal to the whole cone. Another example is that the symmetric space $SL(n,R)/SO(n)$ admits several important equivariant cell decompositions with respect to the arithmetic group $SL(n, Z)$ and hence a cell decomposition of the locally symmetric space $SL(n, Z)/SL(n, R)/SO(n)$. One such decomposition comes from the Minkowski reduction of quadratic forms (or marked lattices). We generalize the Minkowski reduction to marked hyperbolic Riemann surfaces and obtain a solution to a folklore problem: an intrinsic equivariant cell decomposition of the Teichmuller space $T_g$ with respect to the mapping class groups $Mod_g$, which induces a cell decomposition of the moduli space $M_g$. If time permits, I will also discuss other results on similarities between the two classes of spaces and groups.

Host: Lei Ni

### May 14, 2009

### 4:00 PM

### AP&M 6402

****************************