77 research outputs found
Representation growth and representation zeta functions of groups
We give a short introduction to the subject of representation growth and
representation zeta functions of groups, omitting all proofs. Our focus is on
results which are relevant to the study of arithmetic groups in semisimple
algebraic groups, such as the special linear group of degree n over the ring of
integers. In the last two sections we state several results which were recently
obtained in joint work with N. Avni, U. Onn and C. Voll.Comment: 14 pages, submitted to Note di Matematica, survey based on a
conference tal
Igusa-type functions associated to finite formed spaces and their functional equations
We study symmetries enjoyed by the polynomials enumerating non-degenerate
flags in finite vector spaces, equipped with a non-degenerate alternating
bilinear, hermitian or quadratic form. To this end we introduce Igusa-type
rational functions encoding these polynomials and prove that they satisfy
certain functional equations. Some of our results are achieved by expressing
the polynomials in question in terms of what we call parabolic length functions
on Coxeter groups of type . While our treatment of the orthogonal case
exploits combinatorial properties of integer compositions and their
refinements, we formulate a precise conjecture how in this situation, too, the
polynomials may be described in terms of parabolic length functions.Comment: Slightly revised version, to appear in Trans. Amer. Math. Soc
On w-maximal groups
Let be a word, i.e. an element of the free group on generators . The verbal subgroup of
a group is the subgroup generated by the set of all -values in . We say that a (finite)
group is -maximal if for all proper subgroups
of and that is hereditarily -maximal if every subgroup of is
-maximal. In this text we study -maximal and hereditarily -maximal
(finite) groups.Comment: 15 page
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