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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

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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" ($b$), "...
Sebastien Palcoux's user avatar
21 votes
0 answers
473 views

Are braid groups known to not be linear over $\mathbb{Z}$?

$\DeclareMathOperator\GL{GL}$It is known that every braid group $B_n$ embeds as a subgroup of $\GL_m(\mathbb{Z}[q^{\pm 1},t^{\pm 1}])$, where $m=n(n-1)/2$ (see Krammer - Braid groups are linear). This ...
Matt Zaremsky's user avatar
19 votes
0 answers
782 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 ...
Lee Mosher's user avatar
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18 votes
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734 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 ...
Pablo's user avatar
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18 votes
0 answers
477 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 $F_2$. ...
Igor Pak's user avatar
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17 votes
0 answers
255 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 ...
Tomek Odrzygozdz's user avatar
17 votes
0 answers
449 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 ...
Dan Sălăjan's user avatar
16 votes
0 answers
362 views

Does every infinite, connected, locally finite, vertex-transitive graph have a leafless spanning tree?

My question is Let $G$ be an infinite, connected, locally finite, vertex-transitive graph. Must $G$ have the following substructures? i) a leafless spanning tree; ii) a spanning forest consisting ...
Agelos's user avatar
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13 votes
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223 views

Are there free and discrete subgroups of $\mathrm{SL}_2(\mathbb{R}) \times \mathrm{SL}_2(\mathbb{R})$ that are not Schottky on any factor?

$\DeclareMathOperator\SL{SL}$Does there exist a free and discrete subgroup $\Gamma < \SL_2(\mathbb{R}) \times \SL_2(\mathbb{R})$ such that neither $\pi_1(\Gamma)$ nor $\pi_2(\Gamma)$ is free and ...
Ilia Smilga's user avatar
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13 votes
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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 ...
Derek Holt's user avatar
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12 votes
0 answers
479 views

Writing an element of a free product of $C_2$'s as a product of order-$2$ elements

My question is simple: Suppose that $G$ is isomorphic to the free product of finitely many copies of $C_2$. Is it true that any element $g \in G$ can be written as a product $g = s_1 \dotsm s_m$ such ...
Jeff Yelton's user avatar
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12 votes
0 answers
373 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$^*$...
tattwamasi amrutam's user avatar
12 votes
0 answers
397 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 ...
Benjamin Steinberg's user avatar
11 votes
0 answers
345 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 ...
Robbie Lyman's user avatar
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11 votes
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379 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 ...
YCC's user avatar
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11 votes
0 answers
434 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 ...
Nora's user avatar
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11 votes
0 answers
734 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 false (...
Sebastien Palcoux's user avatar
11 votes
0 answers
679 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 ...
user avatar
10 votes
0 answers
223 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) ...
Yellow Pig's user avatar
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10 votes
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351 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 $A_{12} \in \Sigma_b$ ...
Francesco Polizzi's user avatar
10 votes
0 answers
455 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,\...
seldom seen's user avatar
10 votes
0 answers
214 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^{-...
Harry Reed's user avatar
10 votes
0 answers
458 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 ...
scott spencer's user avatar
10 votes
1 answer
331 views

A subgroup of corank 1 in a free group contains a primitive element?

Let $F$ be the free group on $\{x_i\}_{i=1}^\infty$, and let $H \leq F$ be a subgroup with $\langle H \cup \{x_1\} \rangle = F$. Must there be a free basis $B$ of $F$ for which $B \cap H \neq \...
Pablo's user avatar
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9 votes
0 answers
369 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 ...
Stefan Witzel's user avatar
9 votes
0 answers
310 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'...
Sebastien Palcoux's user avatar
9 votes
0 answers
176 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, ...
Michal Ferov's user avatar
9 votes
0 answers
269 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 $...
H1ghfiv3's user avatar
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9 votes
0 answers
487 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 $...
Xiaolei Wu's user avatar
  • 1,598
9 votes
0 answers
556 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 ...
Derek Holt's user avatar
  • 37.4k
8 votes
0 answers
252 views

In search of a quick proof that groups acting freely on $\mathbb R$-trees are linear

Many years ago I had the idea to use non-standard analysis to prove that a group acting freely on an $\mathbb R$-tree must be linear. The heuristic went like this: A non-standard model $G^*$ of the ...
Peter Kropholler's user avatar
8 votes
0 answers
432 views

The figure eight knot complement in $S^3$

Recently I have been going through the book Hyperbolic Knot Theory by Jessica Purcell. Exercise 5.4 (on page 101) gives us a presentation of the fundamental group of $S^3 - K$ where $K$ is the figure-...
T ghosh's user avatar
  • 111
8 votes
0 answers
228 views

Coarse quotient maps

Interesting connections and analogies have been observed between non-linear geometry of Banach spaces and coarse geometry. In the former subject, people have investigated the notion of uniform (or ...
Narutaka OZAWA's user avatar
8 votes
0 answers
186 views

Uniform amenability at infinity

Let's recall that a group $G$ is amenable if for any finite subset $E\subset G$ and any $\epsilon>0$ there is a finite subset $F\subset G$ such that $$\max_{s\in E} |s F \mathbin{\triangle} F| \le \...
Narutaka OZAWA's user avatar
8 votes
0 answers
168 views

Do compact universal covers have concentration of measure phenomenon?

$\DeclareMathOperator\vol{vol}\DeclareMathOperator\diam{diam}$I have a sequence of compact Riemannian manifolds $M_n$ with $\diam (M_n) \to 0$ and finite fundamental groups $\pi_1 (M_n)$ so that their ...
Sergio Zamora's user avatar
8 votes
0 answers
166 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 ...
Matt Zaremsky's user avatar
8 votes
0 answers
149 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 ...
vladkvankov's user avatar
8 votes
0 answers
268 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}])...
user8253417's user avatar
8 votes
0 answers
274 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 ...
ADL's user avatar
  • 2,821
8 votes
0 answers
116 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}(...
user avatar
8 votes
0 answers
189 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 ...
Denis Osin's user avatar
  • 2,648
8 votes
0 answers
323 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 ...
Timm von Puttkamer's user avatar
8 votes
0 answers
185 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 ...
ARG's user avatar
  • 4,432
8 votes
0 answers
343 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 $(\...
Jean Raimbault's user avatar
8 votes
0 answers
298 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\...
Benjamin Steinberg's user avatar
7 votes
0 answers
220 views

Is there a Cayley graph with end space infinite and discrete?

A Cayley graph of a finitely generated group must be locally finite, and we know end spaces of locally finite graphs must be compact - so we can't have an infinite and discrete end space in this ...
violeta's user avatar
  • 407
7 votes
0 answers
120 views

Normal subgroups of pure braid groups stable under strand bifurcation

$\DeclareMathOperator\PB{PB}\DeclareMathOperator\B{B}$Let $\PB_n$ be the $n$-strand pure braid group. For each $1\le k\le n$, let $\kappa_k^n \colon \PB_n \to \PB_{n+1}$ be the monomorphism that takes ...
Matt Zaremsky's user avatar
7 votes
0 answers
193 views

Reduced group C*-algebra $C^*_r(\mathbb{Z}/2*\mathbb{Z}/2)$: norm of specific elements

Consider the free product of $\mathbb{Z}/2$ with itself with generators $$ \mathbb{Z}/2*\mathbb{Z}/2=\langle u,v\mid u^2=1=v^2\rangle $$ and regard its group $C^*$-algebra $$ C^*(\mathbb{Z}/2*\mathbb{...
C-star-W-star's user avatar
7 votes
0 answers
140 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 ...
ARG's user avatar
  • 4,432
7 votes
0 answers
420 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{...
Francesco Polizzi's user avatar

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