3
votes
0answers
62 views

Cancellations in products of two elements of a hyperbolic group

Let $G$ be a non-abelian free group with the standard generating set and the corresponding word metric. If we take two elements $g,h\in G$ and compute their product $gh$, some letters might cancel, ...
0
votes
0answers
54 views

Suitable references for Zappa-Szep products of groups

Can anyone provide references for the following: isoperimetric (Dehn) functions of semi-direct products of groups and, more generally, of Zappa-Szep products of groups?
4
votes
1answer
205 views

Subgroups of Gromov's hyperbolic groups

It's known that subgroups of Gromov's hyperbolic groups are not necessarily hyperbolic. Is there any counter-example when the quotient is Abelian. More precisely, let $G$ be a Gromov's hyperbolic ...
4
votes
1answer
183 views

Boundaries of relatively hyperbolic groups

When the interior of an n-manifold $M$ has a pinched negative curvature metric of finite volume, then its fundamental group $\Gamma=\pi_1M$ is relatively hyperbolic relative to the parabolic groups ...
-1
votes
1answer
128 views

Homeomorphism of the punctured sphere which fixes an essential Jordan curve

$\phi$ is a homeomorphism from the 2-sphere to itself which represents an element of $PMCG(S^2,A)$ (we also denote it by $\phi$), where $A$ is a finite set of $S^2$. $\gamma$ is an essential Jordan ...
4
votes
1answer
228 views

What is known about maximal free subgroups of surface groups?

Let $\Gamma_g=< a_1,...,a_g,b_1,...,b_g | \prod_{i=1}^g [a_i,b_i]>$ (a surface group). What is known about maximal free subgroups of $\Gamma_g$ for $g>1.$ (I.e. free subgroups which are not ...
2
votes
0answers
70 views

Characterizations of product groups under quasi-isometry

This is a follow-up question to Quasi-isometric rigidity of certain products of groups. It can be moved to a comment if it is too trivial. If all the asymptotic cones of a finitely generated group ...
6
votes
0answers
111 views

Is there an analogue of Mostow-Palais equivariant embedding theorem for noncompact groups

Let $M$ be a (Hausdorff) smooth compact manifold and $G$ a Lie group acting smoothly on $M$. If $G$ is compact then, by Mostow-Palais theorem, there exists an equivariant smooth embedding $M\to ...
2
votes
1answer
112 views

Density of ends of long words in a hyperbolic group

Let $G$ be a Gromov hyperbolic group with a generating set $S$. For each $g \in G$, let $\xi_g$ be the point in the ideal boundary corresponding to the sequence $(g^n)$. Let $l_S(g)$ be the word ...
5
votes
0answers
119 views

Uniform incentre of collection of quasi convex subspaces in hyperbolic spaces

I'm reading a paper of Wise on cubulations and the following fact is used: Let $H$ be a quasi-convex subgroup of a $\delta$-hyperbolic group and let $H_i, i\in I$ be a finite family of translates of ...
3
votes
3answers
194 views

Mechanisms generating free subgroups of Artin braid groups

Within my limited experience, I have only known free groups to occur through two mechanisms: as fundamental groups of trees (graphs) and ping-pong. And sometimes only through one way: the fact that ...
3
votes
3answers
217 views

Domination of length functions of trees with equal covolume

(This is a reformulation of an earlier unanswered question. I would like to thank Ian Agol for pointing out to me Walter Parry's characterization of hyperbolic translation length functions.) Let $G$ ...
0
votes
1answer
139 views

discrete subgroups of the isometries of a product

Suppose $X_1$ and $X_2$ are two nice metric spaces, e.g. two Riemannian manifolds, and let $G_i=Isom(X_i)$. Then $G_1\times G_2\subset Isom(X_1\times X_2)$. Suppose $X_1\times X_2$ is not compact and ...
3
votes
1answer
186 views

Symmetries of the standard probability space

The standard probability space $(I, \mathcal B, \lambda)$ consists of the interval $I = [0,1]$, its Borel $\sigma$-algebra $\mathcal B := \mathcal B(I)$ and Lebesgue measure $\lambda$. In ...
6
votes
0answers
236 views

Does a finitely generated aspherical group have an aspherical presentation with a finite generating set?

Let $G$ be a finitely generated group. Suppose $G$ has an aspherical presentation with a countably infinite generating set. Does $G$ have an aspherical presentation with a finite generating set? Here ...
4
votes
1answer
188 views

Interesting families of groups as group extensions

Let me start this question with an example that hopefully makes clear what I am looking for: A discrete subgroup $G$ of the group of euclidean isometries of $\mathbb{R}^d$ is called a ...
7
votes
1answer
295 views

Analogues of the curve complex for Out(F)

Let $F$ be a finitely generated free group. Question: Is there an authoritative survey of analogues of the curve complex for $\mathrm{Out}(F)$? If not, as seems likely, would a passing expert be ...
8
votes
0answers
124 views

Naive Reidemeister-Schreier for $\mathbb Z$ quotients

I have a question about a "standard" variant of the Reidemeister-Schreier algorithm used by topologists when manipulating manifolds they either know or suspect are fibre-bundles over $S^1$. Say you ...
4
votes
1answer
288 views

Can the integer Heisenberg group be cubulated?

I've been interested in abstract polyhedral decompositions of 3-manifolds for a long time. One thing I've tried to do a lot is to get nice polyhedral decompositions of manifolds with Nil geometry. It ...
3
votes
1answer
153 views

Finite generation of the commutator subgroup of the pure braid group

Let $PB_n$ be the pure braid group on $n$ strands. The group $PB_n$ has every conceivable finiteness property. Also, it has a large abelianization. My question is whether the commutator subgroup ...
12
votes
2answers
313 views

Semidirect product decomposition of the Borromean rings group

Let $X=S^3\setminus B$ be the link complement of the Borromean rings. Then $G=\pi_1(X)$ has a presentation of the form $$ G = \langle \; a,b,c \mid [a,[b^{-1},c]],\; [b,[c^{-1},a]], \; ...
5
votes
0answers
223 views

Quotient of 3-sphere by binary octahedral group?

Consider the Lie group $Spin(3)$, which can be thought of geometrically as the 3-sphere (e.g., it can be represented by the collection of unit quaternions). The quotient $Spin(3)/\pm I$ yields the ...
5
votes
1answer
208 views

What is an interpretation of the relation in the cohomology of the pure braid groups?

In 1968, Arnol'd proved that the integral cohomology of the pure braid group $P_n$ is isomorphic to the exterior algebra generated by the collection of degree-one classes $\omega_{i,j}\ (1 \le i < ...
4
votes
1answer
164 views

Periodic automorphisms of free groups and surface homeomorphisms

Denote by $F_n$ the free group of rank $n$. We say that an automorphism $\phi\in Aut(F_n)$ is geometric if there exists a surface with boundary $M$ and a homeomorphism $h\colon M\to M$ such that $h$ ...
9
votes
1answer
455 views

Why isn't $\langle x,y,z|xyzx^{-1}y^{-1}z^{-1}\rangle$ a hyperbolic surface group?

The group mentioned in the title, $\langle x,y,z|xyzx^{-1}y^{-1}z^{-1}=1\rangle$, is in between the torus fundamental group $\langle x,y|xyx^{-1}y^{-1}=1\rangle$ and the two-holed torus fundamental ...
4
votes
2answers
237 views

Do quasi convex hyperbolic subgroups remain quasi convex after adding redundant generators?

We know now that hyperbolic 3-manifolds virtually embed into right-angled Artin groups as quasiconvex subgroups. Also, quasiconvexity depends on the generating set. I have been constructing a space ...
4
votes
2answers
397 views

Are virtual cubulated groups cubulated?

Suppose $G$ has a finite index subgroup $N$ such that $N$ acts properly and cocompactly on a CAT(0)-cube complex. Does $G$ also act properly and cocompactly on a CAT(0)-cube complex? Edit: After ...
0
votes
1answer
140 views

Intersections of subgroups of surface groups [closed]

Let $\mathcal{S}_g$ denote the fundamental group of an oriented surface of genus $g\ge 2$. Does $\mathcal{S}_g$ contain subgroups $A$ and $B$ of finite index such that $A\cap B = \lbrace e\rbrace$?
5
votes
1answer
201 views

QVH characterization of virtually special groups

Agol's recent VHC paper gave a characterization of virtually special groups in terms of being $\mathcal{QVH}$. He remarks that this may be taken as the defining property of virtually special groups ...
7
votes
1answer
206 views

Does this group act geometrically on a Median space?

Let $G$ be the semidirect product of $\mathbb{Z}^2$ with $\mathbb{Z}/6$ where $\mathbb{Z}/6$ acts by the order 6 element of $SL_2(\mathbb{Z})$. We can think of this group as the group of order ...
11
votes
1answer
228 views

Walls of CAT(0) cube complex sufficiently far apart implies intersection of stabilizers finite

I was reading through Agol's paper on the Virtual Haken Conjecture and I came across a claim whose proof I am after. It seems to boil down to the following claim about the hyperplanes and their ...
4
votes
2answers
263 views

Complexification or 'real'ization of Mapping Class group.

So is there a complexification or 'real'ization of the mapping class group or can it be realised as a lattice in some lie group. like $PSL(2, \mathbb Z)$ in $PSL(2, \mathbb R)$. for g=1 this certainly ...
10
votes
2answers
336 views

Invariant free factor of a free group

Let $F_n=F\ast F'$ be a free splitting of the free group $F_n$ and $\phi\in Aut(F_n)$. The free factor $F$ is said to be invariant under $\phi$ if $\phi(F)\subseteq F$. I recently wondered if this ...
4
votes
3answers
261 views

General properties of free-by-cyclic groups

I admit this is a very broad question, but I am looking for general properties of [finitely generated free]-by-[infinite cyclic] groups. More precisely, what are some properties that the groups ...
1
vote
1answer
174 views

Non-trivial action of $SL_n(\mathbb{Z} )$ on a simplicial tree

A group $G$ has Serre's property $FA$ if any isometric action of $G$ on a simplicial tree has a global fixed point. Let $n\geq 3$. It is well-known that $SL_n(\mathbb{Z} )$ has property $FA$. Now my ...
1
vote
1answer
112 views

How to judge a manifold generated by Coxeter system smooth?

For the definition of Coxeter System, you can see: http://en.wikipedia.org/wiki/Coxeter_group Given a chamber Q, given a Coxeter System $(\Gamma,V)$, we can defined a set M by the following way: ...
6
votes
3answers
500 views

Judging whether a finitely presented group is a 3-manifold group?

Given a finitely presented group $G$, how many necessary conditions do people know for $G$ to be isomorphic to the fundamental group of some closed connected 3-manifold? (e.g. residually finite)
9
votes
1answer
273 views

Algorithm to test for discrete or quasi-Fuchsian subgroups of PSL(2,C)

Let $\Gamma = \pi_1(S)$ denote the fundamental group of a compact surface $S$ of genus $g>1$. Given a representation $\rho : \Gamma \to \mathrm{PSL}(2,\mathbb{C})$, specified by matrix ...
2
votes
2answers
220 views

Combination theorems for discrete subgroups of isometry groups

Maskit's combination theorem says: if $M=M_1\cup_\Sigma M_2$ is a union of hyperbolic 3-manifolds $M_1=\Gamma_1\backslash H^3, M_2=\Gamma_2\backslash H^3$ along a surface $\Sigma$, and if the limit ...
4
votes
2answers
309 views

Right Angled Artin Group Reference request

The following should be true: every normal subgroup of a non-Abelian Right Angled Artin Group should contain a free group on two generators. Is there a standard reference one can cite for this?
0
votes
1answer
345 views

Are there a surface group and its two isomorphic subgroups which cannot be transferred to each other under any automorphism of the mother group? [closed]

My question is: Are there $G$, which is a fundamental group of a surface or a 2-orbifal, and its two finite index subgroups $H_1$, $H_2$ satisfying the following condition: $H_1$ is isomorphic to ...
20
votes
2answers
563 views

The image of the point-pushing group in the hyperelliptic representation of the braid group

Let $B_{2g+1}$ be the Artin braid group on $2g+1$ strands. There is a symplectic representation $\rho: B_{2g+1} \rightarrow Sp_{2g}(\mathbf{Z})$ called the "hyperelliptic representation," which ...
11
votes
1answer
348 views

Groups with finitely generated center

Does every group with a finite classifying space have finitely generated center? Remarks: If $G$ is a finitely generated group with infinitely generated center $Z(G)$, then the quotient $G/Z(G)$ ...
3
votes
0answers
302 views

Galois groups and braid groups [closed]

Braid group can be viewed as a symmetry group with a "one more dimension to pass through". Is there any "Galois theory", where the braid groups plays analoguos role as a symmetry groups in a native ...
7
votes
4answers
369 views

Examples of acylindrical 3-manifolds

Let $C$ be the compact cylinder $S^1\times [0,1]$. A 3-manifold $M$ with incompressible boundary is called acylindrical if every map $(C,\partial C)\to (M,\partial M)$ that sends the components of ...
5
votes
1answer
228 views

Normal generation of Torelli

The only normal generators of the Torelli group of a closed surface I can find is Powell's 1977 paper (where the presentation is a bit complicated and given essentially without proof). Is there any ...
4
votes
1answer
307 views

Does the fundamental group of a surface have rigid subgroups?

Suppose $\Gamma$ is the fundamental group of a closed, oriented surface $S$. Let $B$ be a finitely generated, infinite index subgroup of $\Gamma$, and let $\Gamma_B$ be the compact core of the ...
6
votes
1answer
312 views

Periodic automorphisms of free groups

Hi, I am troubled with the following question: Does there exist a finite order automorphism of a free group, $f\in Aut(F)$, such that it fixes no non trivial conjugacy class and no non trivial ...
6
votes
2answers
230 views

equivariant cohomology of the complement to the arrangment $\cup_{i\neq j}overrightarrow{x_i} = overrightarrow{x_j}$?

Let $V=\mathbb{R}^d$ be a $d$-dimensional (Euclidian) vector space over real numbers. Let $G=SO(V)$ be a compact Lie group of linear orthogonal transformations of $V$. Let $Conf_n(V)$ be the space of ...
5
votes
1answer
183 views

Finite index subgroups of the mapping class group with geometric meaning

I have got a question that is perhaps not precise in a mathematical sense. Is there a classification of all coverings of the moduli space of Riemann surfaces which are moduli spaces themselves, that ...