15. ÖMG-Kongress
Jahrestagung der Deutschen Mathematikervereinigung

16. bis 22. September 2001 in Wien

Sektion 1 - Algebra
Dienstag, 18. September 2001, 15.30, Hörsaal 21

 

Trace polynomials of words in SL(2)

Dietrich Burde, Universität Düsseldorf (Koautor: Fritz Grunewald)

 

For $A,B\in SL_2(C)$ and words

$$W = A^{n_1}B^{m_1}\cdots A^{n_t}B^{m_t}$$

$$W' = A^{r_1}B^{s_1}\cdots A^{r_{t'}}B^{s_{t'}}$$

the trace problem is to determine under what conditions $W$ and $W'$ have the same trace for all $A,B\in SL_2(C)$. This question arises from the study of the length spectrum of a Riemann surface, and hence of the eigenvalues of the Laplace operator. The behaviour of these eigenvalues are still mysterious. Depending on the Riemann surface beeing arithmetic or non-arithmetic the eigenvalues of the Laplacian appear to obey two distinct statistical laws: Poissonian in one case, GOE (Gauss orthogonal ensemble) in the other case. We formulate a new conjecture for the trace problem and prove some special cases.

E-Mail:	dburde@math.uni-duesseldorf.de
Homepage:	reh.math.uni-duesseldorf.de/~dburde

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