##### Department of Mathematics,

University of California San Diego

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### Special Algebra

## Laurent Bartholdi

#### University of California, Berkeley

## Algebras and Growth in Branch Groups

##### Abstract:

We compute the structure of the Lie algebras associated to twoexamples of branch groups, and show that one has finite width whilethe other, the \"Gupta-Sidki group\", has unbounded width(Corollary~ ef{cor:gamma:rk}). This answers a question by Sidki.We then draw a general result relating the growth of a branch group,of its Lie algebra, of its graded group ring, and of a naturalhomogeneous space we call emph{parabolic space}, namely thequotient of the group by the stabilizer of an infinite ray. Thegrowth of the group is bounded from below by the growth of itsgraded group ring, which connects to the growth of the Lie algebraby a product-sum formula, and the growth of the parabolic space isbounded from below by the growth of the Lie algebraFinally we use this information to explicitly describe the normalsubgroups of $G$, the \"Grigorchuk group\". All normal subgroupsare characteristic, and the number $b_n$ of normal subgroups of$G$ of index $2^n$ is odd and satisfies${limsup,liminf}b_n/n^{log_2(3)}={5^{log_2(3)},frac29}$.

Host:

### October 28, 2002

### 2:01 PM

### AP&M 7321

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