# Questions tagged [geometric-group-theory]

Large scale properties of groups; growth functions; Dehn functions; small cancellation properties; hyperbolicity and CAT(0); actions and representations; combinatorial group theory; presentations

241
questions with no upvoted or accepted answers

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### Is this representation of Go (game) irreducible?

This post is freely inspired by the basic rules of Go (game), usually played on a $19 \times 19$ grid graph.
Consider the $\mathbb{Z}^2$ grid. We can assign to each vertex a state "black" ($b$), "...

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736 views

### Reference request: Parallel processor theorem of William Thurston

Sometime in the 1980's or 1990's, Bill Thurston proved a theorem regarding the existence of a universal parallel processing machine, using a certain class for such machines having finite deterministic ...

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431 views

### Linear groups which don't contain products of free groups

Let $G \subset GL(n, \Bbb Z)$ be a f.g. linear group. The Tits alternative says that $G$ is either virtually solvable (i.e. has a solvable subgroup of finite index), or contains a free group $F_2$. ...

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653 views

### How boundedly generated is $SL_3(\mathbb{Z})$?

The group $G = \mathrm{SL}_3(\mathbb{Z})$ is known to be boundedly generated, that is, there exists some $m \in \mathbb{N}$, and $g_1, \dots, g_m \in G$ such that we have the following equality of ...

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165 views

### Is there a simple group that is torsion-free, type $\textrm{F}_\infty$, and infinite dimensional?

Does there exist an example of a group that is:
Simple,
Torsion-free,
Of type $\textrm{F}_\infty$, and
Infinite dimensional (meaning of infinite cohomological dimension)?
Thompson's group $F$ has ...

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297 views

### Finitely generated groups with Hölder-exotic space of ends?

The space of ends of a finitely generated group is always homeomorphic to 0, 1, 2 points, or a Cantor set, and in which of these 4 cases it falls is governed by Stallings' characterization (wikipedia ...

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219 views

### Approximation of the effective resistance on Cayley graph

Let $\Gamma$ be a finitely generated group, and denote by $G$ the Cayley graph of $\Gamma$. Denote by $d_R$ the resistance distance metric on this graph. The resistance distance metric between the ...

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415 views

### Splay trees and Thompson's group $F$

( I apologize for only indicating some easy to find references, but new users are not allowed to link more than five). This is very speculative, but:
Question: Is there a reformulation of the Dynamic ...

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276 views

### Bounding the lengths of the conjugators in the word problem for finite group presentations

Let $G = \langle X \mid R \rangle$ be a group defined by a finite presentation, and let $F$ be the free group on $X$. If $w \in F$ represents the identity in $G$, then $w$ is equal in $F$ to (the free ...

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277 views

### Does Thompson's group $V$ have property AP?

Property AP: A discrete group $\Gamma$ has property AP (Approximation Property) if there exists a net $(\phi_i)_{i \in I}$ of finitely supported functions on $\Gamma$ such that $\phi_i \to 1 $ weak$^*$...

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374 views

### What is an example of a word hyperbolic group without a finite complete rewriting system?

I believe that it was an open question back when I was a graduate student whether every word hyperbolic group admits a finite complete (=Church-Rosser=Noetherian+confluent) rewriting system for some ...

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324 views

### Amalgamated product of automatic groups

In Gersten's "Problems on Automatic Groups", Problem 14, he asks the following question: Let $G=A\ast_{C}B$ where $A$ and $B$ are automatic and $C$ is infinite cyclic. Is $G$ automatic?
Is this ...

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350 views

### Other than the Higman group, what other candidates do we have for non-sofic groups?

I know that the Higman group is the most widely studied candidate right now, but what are the others? For example, is (are) Thompson's group(s) sofic? And what about the Burger-Mozes groups? I haven't ...

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635 views

### Definition of a uniformly bounded dual of a group

The unitary dual of a group $G$ is the set of equivalence classes of irreducible unitary representations of $G$ with the Fell topology. (This topology is defined using convergence of positive definite ...

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416 views

### Is there a one-relator circle-packing theorem?

Let $X_w$ be the presentation complex of a one-relator group $\langle x_1,\dotsc,x_n\mid w\rangle$, with $w$ cyclically reduced, i.e., $X_w=R\cup_w D$, with $R$ the rose with $n$ petals labeled $x_1,\...

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169 views

### 2-generator subgroups of an Artin group of small type

Suppose I have an Artin group $G$ of small-type, meaning that the generators either commute or braid. E.g a braid group. Take two generators $g, h$ and arbitrary conjugates of these generators $xgx^{-...

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612 views

### Uniquely geodesic groups

Definition : A group is CAT(0) if it acts properly, cocompactly and isometrically on a CAT(0) space.
Examples : see this blog.
Remark : A CAT(0) space is uniquely geodesic, but the converse is ...

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431 views

### is a group $G$, that admits finite $k(G, 1)$ and has no Baumslag-Solitar subgroups, necessarily hyperbolic?

This is the first question asked in Bestvina's article "Questions in Geometric Group Theory". Does anyone know if there has been any progress made on this problem? Is the question answered if $G$ is ...

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107 views

### Artin groups of type $D_n$ as mapping class groups?

According to Allcock (Braid Pictures for Artin groups, https://arxiv.org/abs/math/9907194), the Artin group $A(D_n$) of type $D_n$ may be realized as an index 2 subgroup of the orbifold fundamental ...

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258 views

### Infinite-dimensional torsion-free $F_\infty$-group not containing $F$

Is there an example of a group $G$ that has the properties
the cohomological dimension of $G$ is infinite: $\operatorname{cd}(G) = \infty$,
$G$ is torsion-free,
$G$ is of type $F_\infty$,
$G$ does ...

**9**

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264 views

### Finite quotients of surface braid groups

Let $\Sigma_b$ be a closed orientable surface of genus $b \geq 2$, and denote by $\mathsf{P}_2(\Sigma_b)$ the pure braid group with two strands on $\Sigma_b$.
There is a braid $A_{12} \in \Sigma_b$ ...

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235 views

### Status of questions in “Group Actions on $\mathbb{R}$-trees”?

Culler and Morgan's "Group Actions on $\mathbb{R}$-trees" lists four questions at the end of the introduction. A few have been famously resolved by work of Rips, Bestvina–Feighn and others.
I'm ...

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145 views

### Can every non-inner automorphism of a group with residually finite outer automorphisms be realised as an non-inner automorphism of a finite quotient?

First some motivation: most proofs that show that the group of outer automorphisms is residually finite do not only show that the subgroup of inner automorphisms is closed in the profinite topology, ...

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261 views

### Sharpness of the $1/6$-constant in the Cancellation Theorem

I originally posted this question over at Stackexchange, before realising it is much better suited for Overflow:
Let $\langle \; S \; | \; R \; \rangle$ be a presentation of a group $G$ with a set $...

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385 views

### When does a CAT(0) group contain a rank one isometry

Let $G$ be a CAT(0) group which acts on the CAT(0) space $X$ properly and cocompactly via isometry. Let $g \in G$ be a hyperbolic isometry of $X$. Then $g$ is called $\textbf{rank one}$ if no axis of $...

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467 views

### Group with unsolvable conjugacy problem but solvable conjugacy length?

Could there exist a finitely presented group with unsolvable conjugacy problem, in which it is decidable whether a word over the group generators is a shortest representative of an element in its ...

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105 views

### Amenable automatic groups

Are there any known examples of finitely generated groups that are both amenable and automatic, besides the easy example of virtually abelian groups? Or are there any known restrictions that arise if ...

**8**

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131 views

### Does a rank 1 CAT(0) space with a proper cocompact group action contain a zero width axis?

A geodesic in a proper CAT(0) space is said to be rank 1 if it does not bound a flat half-plane and zero-width if it does not bound a flat strip of any width.
Let $X$ be a geodesically complete CAT(0) ...

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110 views

### Small flag triangulations

In geometric group theory and low-dimensional topology, asking for a triangulation of a specific space to be flag can often be somewhat more cumbersome than just turning a CW-complex into a simplicial ...

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136 views

### Membership problem in general linear group

This is surely a very well known problem, but I could not find an answer on MO or on Google, so here I am.
Given some finitely generated free subgroup $H$ of $\operatorname{GL}_n(\mathbb{Z}[t,t^{-1}])...

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199 views

### Dualizing module for $\operatorname{Aut}(F_n)$

In The complex of free factors of a free group (pdf at Hatcher's page), Hatcher and Vogtmann defined a simplicial complex $FC_n$ called the ``complex of free factors'' of the free group $F_n$. They ...

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177 views

### Subgroups of torsion-free hyperbolic groups versus subgroups of hyperbolic groups

Let $\mathcal{S}$ be the class of finitely presented torsion-free groups which occur as the subgroup of some hyperbolic group (so for every $G\in \mathcal{S}$ there exists a hyperbolic group $H$ such ...

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201 views

### A group of all whose elements are distorted

Does there exist a finitely presented group, not torsion, all of whose infinite-order elements are distorted?
An infinite-order element $g$ of a finitely generated group $G$ is undistorted if there ...

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242 views

### Breuer-Guralnick-Kantor conjecture and infinite 3/2-generated groups

A group $G$ is called $\frac{3}{2}$-generated if every non-trivial element is contained in a generating pair, i.e. $$\forall g \in G \setminus \{e \}, \ \exists g' \in G \text{ such that } \langle g,g'...

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155 views

### Geodesics between boundary points of a hyperbolic space

Let $X$ be a (not necessarily proper) hyperbolic space. Following Gromov, we define the boundary of $X$ as the set of equivalence classes of sequences convergent at infinity. In general, it is not ...

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294 views

### An infinite torsion group $G$ with finite type $K(G,1)$?

There is a famous open problem in group theory that asks:
Does there exist an infinite finitely presented torsion group?
The general belief being that such groups exist. I would like to know ...

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155 views

### Sharp isoperimetry in the discrete Heisenberg group

The exact shape of the set which has the best isoperimetry in the continuous Heisenberg is (from what I know) a difficult open problem. This brought to wonder what is known in the discrete case?
More ...

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307 views

### Polynomial growth without Gromov's theorem

It is a consequence of the theorem of M. Gromov on groups with polynomial growth and a result of P. Pansu(*) that if a finitely generated group $\Gamma = \langle S\rangle,\, S=S^{-1}$ satisfies $(\...

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286 views

### A Magnus theorem in the category of residually finite groups

There is a natural notion of a presentations in the category of residually finite groups. Namely, if $X$ is set and $R$ is a set of words in the free group $FG(X)$ on $X$, then define $G=RF\langle X\...

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103 views

### Can a lacunary hyperbolic group be Liouville?

While discussing with a colleague of a possible use of lacunary hyperbolic elementarily amenable groups introduced by Olshanskii, Osin & Sapir, it occurred to me that I was not aware whether ...

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386 views

### Are these two kernels isomorphic groups?

We have a finitely presented, infinite group $\mathsf{B}$,
coming from a geometric topology problem (it is the quotient of a braid group for a genus 2 surface).
It is generated by elements
\begin{...

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296 views

### How does Outer Space look like without a simplex?

Considering the simplicial structure of Culler and Vogtmanns Outer Space $CV_n$. The question is now:
Let $\Delta \subset CV_n$ be a closed simplex of dimension $3n-4$ or $3n-5$, how does $CV_n \...

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85 views

### Second cohomology of handlebody mapping class group

Let $H$ be a genus $g$ handlebody, let $\mathrm{Mod}(H)$ be its mapping class group. Is the calculation of $H^2(\mathrm{Mod}(H),\mathbb{Z})$ known?
(Let $S$ be the boundary of $H$, then $\mathrm{Mod}(...

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148 views

### Integral of Gromov product

Consider a geodesic, hyperbolic metric space $(X,d)$. For $x,y \in X$, consider $D(x,y)$ the infimum, over all $1$-Lipschitz paths $p:[a,b] \rightarrow X$ from $x$ to $y$, of the integral $\int_a^b e^{...

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211 views

### Hyperbolic group with boundary $S^1$ implies virtually Fuchsian via bounded cohomology?

Question: Is there an approach to $\partial G \cong S^1$ implies virtually Fuchsian using bounded cohomology of $\mathrm{Homeo^+} (S^1)$? If not is there a reason to believe it wouldn't work, or maybe ...

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179 views

### Elements of a group than can achieve all orders in its quotients

Let $G$ be a group. For different purposes I am now interested in a quite natural property that the elements in $G$ may or may not have, and I would like to ask if there is a standard terminology for ...

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146 views

### A version of co-Hopf property for groups

I would like to know if there is a reasonably common name for the following variation of being co-Hopfian for groups:
A group G is called ? if G is not isomorphic to a subgroup of infinite index in G....

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124 views

### Asymptotic hyperbolicity constant of symmetric group

Let $\delta(G)$ be the smallest Gromov-hyperbolicity constant of a finite group $G$ (e.g., for any generating set of $G$, all triangles in the corresponding Cayley graph are $\delta(G)$-thin). Let $...

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285 views

### Let $G$ be a group of finite cohomological dimension. $\mathbb Z$ normal in $G$. Is it true that $cd_{\mathbb Q}(G/\mathbb Z) < cd_{\mathbb Q} (G)$?

$cd_{\mathbb Q}$ is the cohomological dimension on rational coefficients. Notice that if we replace $\mathbb Q$ with $\mathbb Z$ in $cd_{\mathbb Q}$ the statement is certainly false (think about $\...

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147 views

### Pseudo-Surface Groups

Let $G$ be a residually finite group, and suppose that for every finite index subgroup $U$ of $G$ we have:
$$d(U) = 2[G : U] + 2.$$
Is $G$ necessarily a surface group?
Here $d(U)$ is the least ...