Thirteen papers published by Geometry & Topology Publications

[Geometry and Topology]

Eight papers have been published by Algebraic & Geometric Topology

(1) Algebraic & Geometric Topology 13 (2013) 795-816
���On the autonomous metric on the group of
���area-preserving diffeomorphisms of the 2-disc
�����by Michael Brandenbursky and Jarek Kedra
���URL: http://www.msp.warwick.ac.uk/agt/2013/13-02/p024.xhtml
���DOI: 10.2140/agt.2013.13.795

(2) Algebraic & Geometric Topology 13 (2013) 817-829
���The three smallest compact �arithmetic hyperbolic 5-orbifolds
�����by Vincent Emery and Ruth Kellerhals
���URL: http://www.msp.warwick.ac.uk/agt/2013/13-02/p025.xhtml
���DOI: 10.2140/agt.2013.13.817

(3) Algebraic & Geometric Topology 13 (2013) 831-903
���An etale space construction for stacks
�����by David Carchedi
���URL: http://www.msp.warwick.ac.uk/agt/2013/13-02/p026.xhtml
���DOI: 10.2140/agt.2013.13.831

(4) Algebraic & Geometric Topology 13 (2013) 905-925
���Amenable category of three-manifolds
�����by Jose Carlos Gomez-Larranaga, Francisco Gonzalez-Acuna and
Wolfgang Heil
���URL: http://www.msp.warwick.ac.uk/agt/2013/13-02/p027.xhtml
���DOI: 10.2140/agt.2013.13.905

(5) Algebraic & Geometric Topology 13 (2013) 927-958
���The link volume of 3-manifolds
�����by Yo'av Rieck and Yasushi Yamashita
���URL: http://www.msp.warwick.ac.uk/agt/2013/13-02/p028.xhtml
���DOI: 10.2140/agt.2013.13.927

(6) Algebraic & Geometric Topology 13 (2013) 959-999
���Model categories for orthogonal calculus
�����by David Barnes and Peter Oman
���URL: http://www.msp.warwick.ac.uk/agt/2013/13-02/p029.xhtml
���DOI: 10.2140/agt.2013.13.959

(7) Algebraic & Geometric Topology 13 (2013) 1001-1025
���Cofinite hyperbolic Coxeter groups, minimal growth rate and Pisot
numbers
�����by Ruth Kellerhals
���URL: http://www.msp.warwick.ac.uk/agt/2013/13-02/p030.xhtml
���DOI: 10.2140/agt.2013.13.1001

(8) Algebraic & Geometric Topology 13 (2013) 1027-1047
���Topological complexity of motion planning in projective product
spaces
�����by Jesus Gonzalez, Mark Grant, Enrique Torres-Giese and Miguel
Xicotencatl
���URL: http://www.msp.warwick.ac.uk/agt/2013/13-02/p031.xhtml
���DOI: 10.2140/agt.2013.13.1027

Five papers have been published by Geometry & Topology. �Papers
(9)--(12) complete issue 1 of volume 17, whilst paper 13 opens issue 2

(9) Geometry & Topology 17 (2013) 493-530
���Embedability between right-angled Artin groups
�����by Sang-hyun Kim and Thomas Koberda
���URL: http://www.msp.warwick.ac.uk/gt/2013/17-01/p013.xhtml
���DOI: 10.2140/gt.2013.17.493

(10) Geometry & Topology 17 (2013) 531-562
���Spherical subcomplexes of spherical buildings
�����by Bernd Schulz
���URL: http://www.msp.warwick.ac.uk/gt/2013/17-01/p014.xhtml
���DOI: 10.2140/gt.2013.17.531

(11) Geometry & Topology 17 (2013) 563-593
���On the Hopf conjecture with symmetry
�����by Lee Kennard
���URL: http://www.msp.warwick.ac.uk/gt/2013/17-01/p015.xhtml
���DOI: 10.2140/gt.2013.17.563

(12) Geometry & Topology 17 (2013) 595-620
���Cubic differentials and finite volume convex projective surfaces
�����by Yves Benoist and Dominique Hulin
���URL: http://www.msp.warwick.ac.uk/gt/2013/17-01/p016.xhtml
���DOI: 10.2140/gt.2013.17.595

(13) Geometry & Topology 17 (2013) 621-638
���Kervaire invariants and selfcoincidences
�����by Ulrich Koschorke and Duane Randall
���URL: http://www.msp.warwick.ac.uk/gt/2013/17-02/p017.xhtml
���DOI: 10.2140/gt.2013.17.621

Abstracts follow

(1) On the autonomous metric on the group of ��
���area-preserving diffeomorphisms of the 2-disc
�����by Michael Brandenbursky and Jarek Kedra

Let D^2 be the open unit disc in the Euclidean plane and let G :=
Diff(D^2,area) be the group of smooth compactly supported
area-preserving diffeomorphisms of D^2. �For every natural number k we
construct an injective homomorphism Z^k --> BG, which is bi-Lipschitz
with respect to the word metric on Z^k and the autonomous metric on G.
We also show that the space of homogeneous quasimorphisms vanishing on
all autonomous diffeomorphisms in the above group is
infinite-dimensional.


(2) The three smallest compact �arithmetic hyperbolic 5-orbifolds
�����by Vincent Emery and Ruth Kellerhals

We determine the three hyperbolic 5-orbifolds of smallest volume among
compact arithmetic orbifolds, and we identify their fundamental groups
with hyperbolic Coxeter groups.


(3) An etale space construction for stacks
�����by David Carchedi

We generalize the notion of a sheaf of sets over a space to define the
notion of a small stack of groupoids over an etale stack. We then
provide a construction analogous to the etale space construction in
this context, establishing an equivalence of 2-categories between
small stacks over an etale stack and local homeomorphisms over
it. These results hold for a wide variety of types of spaces, for
example, topological spaces, locales, various types of manifolds, and
schemes over a fixed base (where stacks are taken with respect to the
Zariski topology). Along the way, we also prove that the 2-category of
topoi is fully reflective in the 2-category of localic stacks.


(4) Amenable category of three-manifolds
�����by Jose Carlos Gomez-Larranaga, Francisco Gonzalez-Acuna and
Wolfgang Heil

A closed topological n--manifold M^n is of ame--category at most k if
it can be covered by k open subsets such that for each path-component
W of the subsets the image of its fundamental group pi_1(W) -->
pi_1(M^n) is an amenable group. cat_{ame}(M^n) is the smallest number
k such that M^n admits such a covering. For n=3, M^3 has ame--category
at most 4. We characterize all closed 3--manifolds of ame--category 1,
2 and 3.


(5) The link volume of 3-manifolds
�����by Yo'av Rieck and Yasushi Yamashita

We view closed orientable 3-manifolds as covers of S^3 branched over
hyperbolic links. To a cover p : M -> S^3, of degree p and branched
over a hyperbolic link L \subset S^3, we assign the complexity p
Vol(S^3 \ L). �We define an invariant of 3-manifolds, called the link
volume and denoted by LinkVol(M), that assigns to a 3-manifold M the
infimum of the complexities of all possible covers M -> S^3, where the
only constraint is that the branch set is a hyperbolic link. Thus the
link volume measures how efficiently M can be represented as a cover
of S^3.
We study the basic properties of the link volume and related
invariants, in particular observing that for any hyperbolic manifold
M, Vol(M) < LinkVol(M). �We prove a structure theorem that is similar
to (and uses) the celebrated theorem of Jorgensen and Thurston. This
leads us to conjecture that, generically, the link volume of a
hyperbolic 3-manifold is much bigger than its volume.
Finally we prove that the link volumes of the manifolds obtained by Dehn
filling a manifold with boundary tori are linearly bounded above in
terms
of the length of the continued fraction expansion of the filling curves.


(6) Model categories for orthogonal calculus
�����by David Barnes and Peter Oman

We restate the notion of orthogonal calculus in terms of model
categories. �This provides a cleaner set of results and makes the role
of O(n)-equivariance clearer. Thus we develop model structures for the
category of n-polynomial and n-homogeneous functors, along with
Quillen pairs relating them. We then classify n-homogeneous functors,
via a zig-zag of Quillen equivalences, in terms of spectra with an
O(n)-action. This improves upon the classification theorem of
Weiss. As an application, we develop a variant of orthogonal calculus
by replacing topological spaces with orthogonal spectra.


(7) Cofinite hyperbolic Coxeter groups, minimal growth rate and Pisot
numbers
�����by Ruth Kellerhals

By a result of R Meyerhoff, it is known that among all cusped
hyperbolic 3-orbifolds the quotient of H^3 by the tetrahedral Coxeter
group (3,3,6) has minimal volume. �We prove that the group (3,3,6) has
smallest growth rate among all non-cocompact cofinite hyperbolic
Coxeter groups, and that it is as such unique. This result extends to
three dimensions some work of W. Floyd who showed that the Coxeter
triangle group (3,infinity) has minimal growth rate among all
non-cocompact cofinite planar hyperbolic Coxeter groups. �In contrast
to Floyd's result, the growth rate of the tetrahedral group (3,3,6) is
not a Pisot number.


(8) Topological complexity of motion planning in projective product
spaces
�����by Jesus Gonzalez, Mark Grant, Enrique Torres-Giese and Miguel
Xicotencatl

We study Farber's topological complexity (TC) of Davis' projective
product spaces (PPS's). We show that, in many nontrivial instances,
the TC of PPS's coming from at least two sphere factors is (much)
lower than the dimension of the manifold. This is in marked contrast
with the known situation for (usual) real projective spaces for which,
in fact, the Euclidean immersion dimension and TC are two facets of
the same problem. Low TC-values have been observed for infinite
families of nonsimply connected spaces only for H-spaces, for finite
complexes whose fundamental group has cohomological dimension at most
2, and now in this work for infinite families of PPS's. We discuss
general bounds for the TC (and the Lusternik-Schnirelmann category) of
PPS's, and compute these invariants for specific families of such
manifolds. Some of our methods involve the use of an equivariant
version of TC. We also give a characterization of the Euclidean
immersion dimension of PPS's through a generalized concept of axial
maps or, alternatively (in an appendix), nonsingular maps. This gives
an explicit explanation of the known relationship between the
generalized vector field problem and the Euclidean immersion problem
for PPS's.


(9) Embedability between right-angled Artin groups
�����by Sang-hyun Kim and Thomas Koberda

In this article we study the right-angled Artin subgroups of a given
right-angled Artin group. �Starting with a graph Gamma, we produce a
new graph through a purely combinatorial procedure, and call it the
extension graph Gamma^e of Gamma. �We produce a second graph
Gamma^e_k, the clique graph of Gamma^e, by adding an extra vertex for
each complete subgraph of Gamma^e. �We prove that each finite induced
subgraph Lambda of Gamma^e gives rise to an inclusion A(Lambda) -->
A(Gamma). �Conversely, we show that if there is an inclusion A(Lambda)
--> A(Gamma) then Lambda is an induced subgraph of Gamma^e_k. �These
results have a number of corollaries. �Let P_4 denote the path on four
vertices and let C_n denote the cycle of length n. �We prove that
A(P_4) embeds in A(Gamma) if and only if P_4 is an induced subgraph of
Gamma. �We prove that if F is any finite forest then A(F) embeds in
A(P_4). �We recover the first author's result on co-contraction of
graphs, and prove that if Gamma has no triangles and A(Gamma) contains
a copy of A(C_n) for some n at least 5, then Gamma contains a copy of
C_m for some m that is at least 5 and at most n. �We also recover
Kambites' Theorem, which asserts that if A(C_4) embeds in A(Gamma)
then Gamma contains an induced square. �We show that whenever Gamma is
triangle-free and A(Lambda) < A(Gamma) then there is an undistorted
copy of A(Lambda) in A(Gamma). �Finally, we determine precisely when
there is an inclusion A(C_m) --> A(C_n) and show that there is no
``universal'' two--dimensional right-angled Artin group.


(10) Spherical subcomplexes of spherical buildings
�����by Bernd Schulz

Let Delta be a thick, spherical building equipped with its natural
CAT(1) metric and let M be a proper, convex subset of Delta. If M is
open or if M is a closed ball of radius pi/2, then Lambda, the maximal
subcomplex supported by Delta - M, is dim Lambda-spherical and
non-contractible.


(11) On the Hopf conjecture with symmetry
�����by Lee Kennard

The Hopf conjecture states that an even-dimensional, positively curved
Riemannian manifold has positive Euler characteristic. We prove this
conjecture under the additional assumption that a torus acts by
isometries and has dimension bounded from below by a logarithmic
function of the manifold dimension. The main new tool is the action of
the Steenrod algebra on cohomology.


(12) Cubic differentials and finite volume convex projective surfaces
�����by Yves Benoist and Dominique Hulin

We prove that there exists a natural bijection between the set of
finite volume oriented convex projective surfaces with nonabelian
fundamental group and the set of finite volume hyperbolic Riemann
surfaces endowed with a holomorphic cubic differential with poles of
order at most 2 at the cusps.


(13) Kervaire invariants and selfcoincidences
�����by Ulrich Koschorke and Duane Randall

Minimum numbers decide, eg, whether a given map f: S^m --> S^n/G from
a sphere into a spherical space form can be deformed to a map f' such
that f(x) is not equal to f'(x) for all x in S^m. �In this paper we
compare minimum numbers to (geometrically defined) Nielsen numbers
(which are more computable). �In the stable dimension range these
numbers coincide. �But already in the first nonstable range (when
m=2n-2) the Kervaire invariant appears as a decisive additional
obstruction which detects interesting geometric coincidence phenomena.
Similar results (involving, eg, Hopf invariants taken mod 4) are
obtained in the next seven dimension ranges (when 1<m-2n+3 is less
than or equal to 8). �The selfcoincidence context yields also a
precise geometric criterion for the open question whether the Kervaire
invariant vanishes on the 126-stem or not.



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