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
327 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" ($b$), "...
21
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0
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473
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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 ...
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19
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0
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782
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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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18
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734
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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 ...
- 11.3k
18
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0
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477
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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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17
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0
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255
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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 ...
- 395
17
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449
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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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16
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362
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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 ...
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13
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0
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223
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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 ...
- 1,574
13
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289
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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 ...
- 37.4k
12
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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 ...
- 1,298
12
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0
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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$^*$...
12
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0
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397
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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 ...
- 38.6k
11
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0
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345
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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 ...
- 1,996
11
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379
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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 ...
- 525
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434
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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 ...
- 111
11
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734
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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 (...
11
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0
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679
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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 ...
user2412
10
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223
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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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10
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351
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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 $A_{12} \in \Sigma_b$ ...
- 66.3k
10
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0
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455
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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,\...
- 436
10
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0
answers
214
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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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458
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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 ...
- 101
10
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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 \...
- 11.3k
9
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0
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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 ...
- 2,050
9
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0
answers
310
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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'...
9
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0
answers
176
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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, ...
- 191
9
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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 $...
- 1,255
9
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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 $...
- 1,598
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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 ...
- 37.4k
8
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0
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252
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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 ...
- 1,638
8
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0
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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-...
- 111
8
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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 ...
- 10.1k
8
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0
answers
186
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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 \...
- 10.1k
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 ...
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8
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0
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166
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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 ...
- 3,740
8
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0
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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 ...
- 101
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0
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268
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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}])...
- 469
8
votes
0
answers
274
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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 ...
- 2,821
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0
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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}(...
user39380
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 ...
- 2,648
8
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0
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323
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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 ...
8
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0
answers
185
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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 ...
- 4,432
8
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0
answers
343
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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 $(\...
- 3,399
8
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0
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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\...
- 38.6k
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 ...
- 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 ...
- 3,740
7
votes
0
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192
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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{...
- 598
7
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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 ...
- 4,432
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votes
0
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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{...
- 66.3k