## Mathematics## Seminar |

## Rocky Mountain Algebraic Combinatorics Seminar

### Linear actions of $\mathbb{Z}/p \times \mathbb{Z}/p$ on $S^n \times S^n$

Jim Fowler

The Ohio State University

In 1925, Hopf first stated the *spherical space form* asking
for which groups act freely on *S*^{n}. Some fifty years later, Madsen,
Thomas, and Wall proved in 1978 that certain necessary conditions
(discovered by Smith in 1944 and Milnor in 1957) were in fact
sufficient. Easy generalizations of this question, like determining
which groups *G* can act freely on *S*^{n} ×*S*^{n}, are still open. Even for a
fixed group, there is the question of classifying the possible actions.
The situation of *linear* actions of \mathbb*Z*/*p* ×\mathbb*Z*/*p*
on *S*^{n} ×*S*^{n} can partly be understood by relating them to the
easier case of \mathbb*Z*/*p* actions on *S*^{n}, that is, to lens spaces.

### Maximal subgroup growth of some groups

Andrew Kelley

Binghamton University

Let m_{n}(G) denote the number of maximal subgroups of a finitely
generated group G of index n. How do the algebraic/structural
properties of G control the growth rate of m_{n}(G)? Others have
researched the broad picture and described what it means for m_{n}(G)
to be bounded above by a polynomial in n. However, there are only
a few groups whose degree of growth is known. If we restrict to
particularly nice classes of groups however, then asymptotic formulas
(or bounds) can be given. We will focus on metabelian groups, especially
those that are abelian by cyclic. Beyond this, current progress on
virtually abelian groups and Baumslag-Solitar groups may also be mentioned.

Weber 223

4–6 pm

Friday, December 2, 2016

(Refreshments in Weber 117, 3:30–4 pm)

Colorado State University

This is a joint Denver U / UC Boulder / UC Denver / U of Wyoming / CSU seminar that meets biweekly. Anyone interested is welcome to join us at a local restaurant for dinner after the talks.