##### Department of Mathematics,

University of California San Diego

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### Food For Thought Seminar

## Blake Rector

#### UCSD

## Characterization of Solenoidal Groups

##### Abstract:

A topological group $G$ is said to be solenoidal if it contains a dense one-parameter subgroup. That is, there exists a continuous homomorphism from the real numbers into $G$ such that the image is dense. We can obtain information about a topological group by studying its character group, the set of all continuous homomorphisms from $G$ into the circle group $T$. This is analogous to studying the dual of a vector space in functional analysis. We show that a compact group is solenoidal if and only if its character group is topologically isomorphic with a subgroup of the real line with the discrete topology. Along the way, we encounter the$a$-adic numbers, the $a$-adic solenoid, and ultimately a corollary which tells us that all compact solenoids can be expressed in terms of an uncountable product of $a$-adic solenoids.

### February 18, 2010

### 10:00 AM

### AP&M 7321

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