315 research outputs found
Representation Growth of Linear Groups
Let be a group and the number of its -dimensional
irreducible complex representations. We define and study the associated
representation zeta function \calz_\Gamma(s) = \suml^\infty_{n=1}
r_n(\Gamma)n^{-s}. When is an arithmetic group satisfying the
congruence subgroup property then \calz_\Gamma(s) has an ``Euler
factorization". The "factor at infinity" is sometimes called the "Witten zeta
function" counting the rational representations of an algebraic group. For
these we determine precisely the abscissa of convergence. The local factor at a
finite place counts the finite representations of suitable open subgroups
of the associated simple group over the associated local field . Here we
show a surprising dichotomy: if is compact (i.e. anisotropic over
) the abscissa of convergence goes to 0 when goes to infinity, but
for isotropic groups it is bounded away from 0. As a consequence, there is an
unconditional positive lower bound for the abscissa for arbitrary finitely
generated linear groups. We end with some observations and conjectures
regarding the global abscissa
Subgroup growth of lattices in semisimple Lie groups
We give very precise bounds for the congruence subgroup growth of arithmetic
groups. This allows us to determine the subgroup growth of irreducible lattices
of semisimple Lie groups. In the most general case our results depend on the
Generalized Riemann Hypothesis for number fields but we can state the following
unconditional theorem:
Let be a simple Lie group of real rank at least 2, different than
D_4(\bbc), and let be any non-uniform lattice of . Let
denote the number of subgroups of index at most in .
Then the limit exists and equals a constant which depends only on
the Lie type of and can be easily computed from its root system.Comment: 34 page
Dimension expanders
We show that there exists k \in \bbn and 0 < \e \in\bbr such that for
every field of characteristic zero and for every n \in \bbn, there exists
explicitly given linear transformations satisfying
the following:
For every subspace of of dimension less or equal ,
\dim(W+\suml^k_{i=1} T_iW) \ge (1+\e) \dim W. This answers a question of Avi
Wigderson [W]. The case of fields of positive characteristic (and in particular
finite fields) is left open
Proalgebraic crossed modules of quasirational presentations
We introduce the concept of quasirational relation modules for discrete and
pro- presentations of discrete and pro- groups and show that aspherical
presentations and their subpresentations are quasirational. In the pro--case
quasirationality of pro--groups with a single defining relation holds. For
every quasirational (pro-)relation module we construct the so called
-adic rationalization, which is a pro-fd-module
. We provide the isomorphisms
and , where and
stands for continuous prounipotent completions and corresponding
prounipotent presentations correspondingly. We show how
embeds into a sequence of abelian prounipotent
groups. This sequence arises naturally from a certain prounipotent crossed
module, the latter bring concrete examples of proalgebraic homotopy types. The
old-standing open problem of Serre, slightly corrected by Gildenhuys, in its
modern form states that pro--groups with a single defining relation are
aspherical. Our results give a positive feedback to the question of Serre.Comment: This is a corrected version of the paper which appeared in the
Extended Abstracts Spring 2015, Interactions between Representation Theory,
Algebraic Topology and Commutative Algebra, Research Perspectives CRM
Barcelona, Vol.5, 201
Invariable generation and the chebotarev invariant of a finite group
A subset S of a finite group G invariably generates G if G = <hsg(s) j s 2 Si
> for each choice of g(s) 2 G; s 2 S. We give a tight upper bound on the
minimal size of an invariable generating set for an arbitrary finite group G.
In response to a question in [KZ] we also bound the size of a randomly chosen
set of elements of G that is likely to generate G invariably. Along the way we
prove that every finite simple group is invariably generated by two elements.Comment: Improved versio
Presentations: from Kac-Moody groups to profinite and back
We go back and forth between, on the one hand, presentations of arithmetic
and Kac-Moody groups and, on the other hand, presentations of profinite groups,
deducing along the way new results on both
Planar Induced Subgraphs of Sparse Graphs
We show that every graph has an induced pseudoforest of at least
vertices, an induced partial 2-tree of at least vertices, and an
induced planar subgraph of at least vertices. These results are
constructive, implying linear-time algorithms to find the respective induced
subgraphs. We also show that the size of the largest -minor-free graph in
a given graph can sometimes be at most .Comment: Accepted by Graph Drawing 2014. To appear in Journal of Graph
Algorithms and Application
Recommended from our members
Generalized triangle groups, expanders, and a problem of Agol and Wise
Answering a question asked by Agol and Wise, we show that a desired stronger
form of Wise's malnormal special quotient theorem does not hold. The
counterexamples are generalizations of triangle groups, built using the
Ramanujan graphs constructed by Lubotzky--Phillips--Sarnak
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