9 research outputs found
Virtual algebraic fibrations of K\"ahler groups
This paper stems from the observation (arising from work of T. Delzant) that
"most" K\"ahler groups virtually algebraically fiber, i.e. admit a finite index
subgroup that maps onto with finitely generated kernel. For the
remaining ones, the Albanese dimension of all finite index subgroups is at most
one, i.e. they have virtual Albanese dimension . We show that the
existence of algebraic fibrations has implications in the study of coherence
and higher BNSR invariants of the fundamental group of aspherical K\"ahler
surfaces. The class of K\"ahler groups with includes virtual
surface groups. Further examples exist; nonetheless they exhibit a strong
relation with surface groups. In fact, we show that the Green--Lazarsfeld sets
of groups with (virtually) coincide with those of surface groups,
and furthermore that the only virtually RFRS groups with are
virtually surface groups.Comment: Substantial revision, including a change of title to better reflect
the content. To appear in Nagoya Math.
Groups quasi-isometric to RAAG's
We characterize groups quasi-isometric to a right-angled Artin group with
finite outer automorphism group. In particular all such groups admit a
geometric action on a cube complex that has an equivariant "fibering"
over the Davis building of .Comment: Minor corrections in the introduction, acknowledgement adde
Cubulated groups: thickness, relative hyperbolicity, and simplicial boundaries
Let G be a group acting geometrically on a CAT(0) cube complex X. We prove
first that G is hyperbolic relative to the collection P of subgroups if and
only if the simplicial boundary of X is the disjoint union of a nonempty
discrete set, together with a pairwise-disjoint collection of subcomplexes
corresponding, in the appropriate sense, to elements of P. As a special case of
this result is a new proof, in the cubical case, of a Theorem of
Hruska--Kleiner regarding Tits boundaries of relatively hyperbolic CAT(0)
spaces. Second, we relate the existence of cut-points in asymptotic cones of a
cube complex X to boundedness of the 1-skeleton of the boundary of X. We deduce
characterizations of thickness and strong algebraic thickness of a group G
acting properly and cocompactly on the CAT(0) cube complex X in terms of the
structure of, and nature of the G-action on, the boundary of X. Finally, we
construct, for each n,k, infinitely many quasi-isometry types of group G such
that G is strongly algebraically thick of order n, has polynomial divergence of
order n+1, and acts properly and cocompactly on a k-dimensional CAT(0) cube
complex.Comment: Corrections according to referee report. Fixed proof of Theorem 4.3.
To appear in "Groups, Geometry, and Dynamics
Cell complexes, poset topology and the representation theory of algebras arising in algebraic combinatorics and discrete geometry
In recent years it has been noted that a number of combinatorial structures
such as real and complex hyperplane arrangements, interval greedoids, matroids
and oriented matroids have the structure of a finite monoid called a left
regular band. Random walks on the monoid model a number of interesting Markov
chains such as the Tsetlin library and riffle shuffle. The representation
theory of left regular bands then comes into play and has had a major influence
on both the combinatorics and the probability theory associated to such
structures. In a recent paper, the authors established a close connection
between algebraic and combinatorial invariants of a left regular band by
showing that certain homological invariants of the algebra of a left regular
band coincide with the cohomology of order complexes of posets naturally
associated to the left regular band.
The purpose of the present monograph is to further develop and deepen the
connection between left regular bands and poset topology. This allows us to
compute finite projective resolutions of all simple modules of left regular
band algebras over fields and much more. In the process, we are led to define
the class of CW left regular bands as the class of left regular bands whose
associated posets are the face posets of regular CW complexes. Most of the
examples that have arisen in the literature belong to this class. A new and
important class of examples is a left regular band structure on the face poset
of a CAT(0) cube complex. Also, the recently introduced notion of a COM
(complex of oriented matroids or conditional oriented matroid) fits nicely into
our setting and includes CAT(0) cube complexes and certain more general CAT(0)
zonotopal complexes. A fairly complete picture of the representation theory for
CW left regular bands is obtained.Comment: Revised based on referee suggestion
Geometry and combinatorics via right-angled Artin groups
We survey the relationship between the combinatorics and geometry of graphs
and the algebraic structure of right-angled Artin groups. We concentrate on the
defining graph of the right-angled Artin group and on the extension graph
associated to the right-angled Artin group. Additionally, we discuss
connections to geometric group theory and complexity theory. The final version
of this survey will appear in "In the tradition of Thurston, vol.~II",
ed.~K.~Ohshika and A.~Papadopoulos.Comment: 44 page
New Perspectives on the Interplay between Discrete Groups in Low-Dimensional Topology and Arithmetic Lattices
This workshop brought together specialists in areas ranging from arithmetic groups to topological quantum field theory, with common interest in arithmetic aspects of discrete groups arising from topology. The meeting showed significant progress in the field and enhanced the many connections between its subbranches
A survey of the impact of Thurston's work on Knot Theory
This is a survey of the impact of Thurston's work on knot theory, laying
emphasis on the two characteristic features, rigidity and flexibility, of
3-dimensional hyperbolic structures. We also lay emphasis on the role of the
classical invariants, the Alexander polynomial and the homology of finite
branched/unbranched coverings.Comment: 97 pages, 12 figure
Ahlfors circle maps and total reality: from Riemann to Rohlin
This is a prejudiced survey on the Ahlfors (extremal) function and the weaker
{\it circle maps} (Garabedian-Schiffer's translation of "Kreisabbildung"), i.e.
those (branched) maps effecting the conformal representation upon the disc of a
{\it compact bordered Riemann surface}. The theory in question has some
well-known intersection with real algebraic geometry, especially Klein's
ortho-symmetric curves via the paradigm of {\it total reality}. This leads to a
gallery of pictures quite pleasant to visit of which we have attempted to trace
the simplest representatives. This drifted us toward some electrodynamic
motions along real circuits of dividing curves perhaps reminiscent of Kepler's
planetary motions along ellipses. The ultimate origin of circle maps is of
course to be traced back to Riemann's Thesis 1851 as well as his 1857 Nachlass.
Apart from an abrupt claim by Teichm\"uller 1941 that everything is to be found
in Klein (what we failed to assess on printed evidence), the pivotal
contribution belongs to Ahlfors 1950 supplying an existence-proof of circle
maps, as well as an analysis of an allied function-theoretic extremal problem.
Works by Yamada 1978--2001, Gouma 1998 and Coppens 2011 suggest sharper degree
controls than available in Ahlfors' era. Accordingly, our partisan belief is
that much remains to be clarified regarding the foundation and optimal control
of Ahlfors circle maps. The game of sharp estimation may look narrow-minded
"Absch\"atzungsmathematik" alike, yet the philosophical outcome is as usual to
contemplate how conformal and algebraic geometry are fighting together for the
soul of Riemann surfaces. A second part explores the connection with Hilbert's
16th as envisioned by Rohlin 1978.Comment: 675 pages, 199 figures; extended version of the former text (v.1) by
including now Rohlin's theory (v.2