0
votes
0answers
45 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, ...
2
votes
0answers
59 views

Embedding a linearly ordered free monoid into a linearly ordered group

A linearly ordered (shortly, l.o.) monoid is a triple $\mathbb M = (M, \cdot, \le)$ for which $(M, \cdot)$ is a (multiplicatively written) monoid and $\le$ is a total order on $M$ such that $xy < ...
7
votes
4answers
441 views

Conjugation Quandles and… “Quandle-Groups”? From quandles to Groups

This question is already asked MathSE A quandle $(Q,*,/ )$ is a idempotent right-distributive and right invertible structure. 1) $a*a=a$ 2) $(a*b)*c=(a*c)*(b*c)$ 3) $(a*b) ...
1
vote
2answers
200 views

Lattices in general totally disconnected locally compact groups

Besides automorphism groups of trees and buildings, I was wondering if the lattices in general totally disconnected locally compact groups have been studied in the literature? I appreciate if you ...
2
votes
0answers
90 views

An equivariant Hahn Embedding Theorem?

The Hahn Embedding Theorem asserts that for any (linearly) ordered abelian group $\Lambda$, there exists a linearly ordered indexing set $\Omega$ such that $\Lambda$ admits an order-preserving group ...
1
vote
1answer
235 views

Odd subgroup of $\mathrm{GL}(n,\mathbb{Z})$

The group $\mathrm{GL}(n,\mathbb{Z})$ acts on $(\mathbb{Z}/2\mathbb{Z})^n$ by right multiplication (the same kind of things can be done with left action). I denote by $H\subset ...
4
votes
1answer
141 views

Which hyperbolic tilings are Cayley graphs?

I realise the question is easy but after asking to a few people (and never getting a clear answer), I thought it could be instructive to ask it here: Given a regular tiling of the hyperbolic plane is ...
4
votes
1answer
182 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 ...
3
votes
2answers
316 views

A structure of the group of automorphisms of an infinite binary tree

My friend asked me to ask his question here. Where he can find (a paper or a book) containing a complete description (with the proof) of a structure of the group of automorphisms of an infinite binary ...
2
votes
1answer
166 views

Doubly primitive groups with simple socle

The classification of doubly transitive groups with simple socle is known. A good account of such classification can be found for example in this paper: Cameron, Peter J. Finite permutation groups ...
18
votes
2answers
484 views

Reference for the triple covering of A_6

I would like to ask for a reference (book, paper ...) for the following nice construction, which I have found as an exercise in some notes of a course by R. Borcherds. For $n=6$ or $7$ (and only in ...
11
votes
1answer
233 views

Group cochains invariant under the action of the symmetric group

Let $G$ be a finite group and $A$ an abelian group. Recall the cochain groups $$ C^k = \{f: G^k \to A\} $$ and the coboundary map $$ \delta : C^k \to C^{k+1} $$ $$ (\delta f)(g_1, \ldots, g_{k+1}) ...
8
votes
1answer
245 views

Calculations of nonabelian group cohomology of R^n

I am looking at $H^1(\mathbb{R}^n,G)$ where $G$ is a finite 2-group. I'm wondering if such things have been calculated. I'm afraid I can't say I know anything here, past the result that this ...
10
votes
1answer
413 views

When taking the fixed points commutes with taking the orbits

Let $G$ and $H$ be groups, both acting on a set $X$ on the left, in such a way that the two actions commute. (Equivalently, let $G \times H$ act on $X$.) The set $\text{Fix}_H(X)$ of $H$-fixed ...
6
votes
4answers
313 views

Is the conjugacy problem solvable in $Out(F_n)$?

There is a paper of Martin Lustig on his webpage giving a positive answer to the conjugacy problem for the outer automorphism group of the free group $F_n$. On the other hand, there seems not to be a ...
0
votes
1answer
126 views

resources in surjunctive groups

Are there any free available resources on surjunctive groups which are available to say: a graduate level student? A textbook would be fine also. Regards.
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 ...
0
votes
0answers
55 views

Order of non-trivial zeros of an L-function and topological dimension

Let $F$ be a primitive element of the Selberg class of degree $d_{F}>0$, and let's consider the group $G$ of complex isometries of finite order that preserve the critical strip ...
8
votes
1answer
195 views

Orderability of $\langle a,b \mid a^{-1} b^m a = b^n\rangle$ and $\langle a,b \mid a^{-1} b a^m = b^n\rangle$

We say that a group $(A, \cdot)$ is bi-orderable if there exists a total order $\preceq$ on $A$ such that $xz \prec yz$ and $zx \prec zy$ for all $x,y,z \in A$ with $x \prec y$. Let $m,n$ be non-zero ...
6
votes
1answer
436 views

Automorphism group of a finite group

I would like to ask if there exists an explicit description of $\mathrm{Aut}(G)$, the group of automorphisms of a finite group $G$, in particular, when $G$ is abelian. E.g., if $G = ...
8
votes
1answer
499 views

$SL_2(\mathbf{Z},8\mathbf{Z})$ differs from $E_2(\mathbf{Z},8\mathbf{Z})$. Has this result appeared in the literature?

I know a proof that the congruence subgroup $SL_2(\mathbf{Z},8\mathbf{Z})$ differs from its subgroup $E_2(\mathbf{Z},8\mathbf{Z})$, but can't find this fact in the literature. Does anyone know a ...
1
vote
0answers
100 views

Is the automorphism group of a homogeneous (locally finite) tree unimodular?

I have seen somewhere (that I don't remember now) that the (full) automorphism group of a k-regular tree is unimodular. I assume a k-regular tree is the same thing as the homogeneous tree of degree k ...
4
votes
0answers
63 views

Estimates for simple random walks in groups of intermediate growth

I'm looking for references for the rate of escape and return probability for a group of intermediate growth. Let $0<\alpha < 1$. If the volume growth is $\succeq \mathrm{exp}(n^\alpha)$, then ...
7
votes
1answer
294 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 ...
7
votes
2answers
319 views

A question on some computation of group cohomologies

Let $G=H\times J$, where $H\cong J\cong C_2$ (cyclic group of order 2). Let $M \cong \mathbb{Z}$ be a $G$-module via "trivial $H$-action and negation $J$-action". My question is "What are the group ...
6
votes
1answer
184 views

Is there an agreed-upon name for this type of subgroup?

In Embedding theorems for groups, (J. London Math. Soc. 34 1959 465–479.) Neumann and Neumann (NB: this is not the Higman-Neumann-Neumann paper of the same name) make the following definition. ...
2
votes
0answers
86 views

Groups acting on non-locally-finite trees with independence and specified local actions

Suppose I have a biregular tree $T_{m, n}$ (not necessarily locally finite), with distinct cardinal numbers $m, n$, so Aut$(T_{m, n})$ acts on $T_{m, n}$ without inversion. Let $V_m$ be those vertices ...
5
votes
1answer
216 views

growth rate of $\mathbb{Z}^2\rtimes_{\sigma} \mathbb{Z}$?

I am interested in the growth rate of this type of group: $G=\mathbb{Z}^2\rtimes_{\sigma} \mathbb{Z}$, where $\sigma(a)=\begin{pmatrix}x&y\\z&w\end{pmatrix}\in SL_2(\mathbb{Z})$, where $a$ is ...
6
votes
1answer
203 views

Easy argument for “connected simple real rank zero Lie groups are compact”?

Let $G$ be a connected simple Lie group. It is known that if $G$ has real rank zero, then $G$ is compact. Background: every connected (semi)simple Lie group $G$ (with Lie algebra $\mathfrak{g}$) has ...
6
votes
1answer
217 views

Reference for embeddings of reflection groups (related to folding ADE Coxeter graphs?

There are a couple of indirect methods, using Lie theory or Springer's general theory of regular elements in (real, complex) reflection groups, to construct natural embeddings among certain Weyl ...
9
votes
4answers
465 views

Groups as Union of Proper Subgroups: References

There are interesting theorems about groups as union of proper subgroups. The first result in this subject is the theorem of Scorza(1926): "a groups if union of three proper subgroups if and only it ...
4
votes
2answers
170 views

Reference for tetrahedral Coxeter group

Let G be the group with 4 generators, each of order 2, such that the product of any 2, say ab, has order 3 (i.e., ababab=e). That is, this is an infinite reflection group with Coxeter diagram a ...
3
votes
1answer
219 views

Speed of random walks in groups

I've seen some estimates for the decay in $d$ of the probability a SRW makes a distance $d$ in time $n$, but is there any reference for the "speed" of a random walk in a group? I'm interested mostly ...
2
votes
1answer
181 views

Divergence of geodesics in mapping class groups

I'm trying to learn some stuff about divergence of geodesics. Let $\gamma$ be a geodesic in a metric space $X$. The divergence of $\gamma$ is a function $f(r)$ for $r \ge 0$ such that $f(r)$ is the ...
9
votes
1answer
262 views

quasi-homomorphisms of groups

Suppose that $G$ is a group and $d$ is a left-invaraint metric on $G$, e.g., the word metric (provided that $G$ is finitely-generated) or distance function determined by a left-invariant Riemannian ...
9
votes
3answers
480 views

History of profinite groups, when was it first mentioned? What was the original definition?

Searching left me hanging. One of my professors told me the definition using the topological properties was the first one but I cannot find any resources. Is that true? If not, how was it originally ...
2
votes
1answer
129 views

Subgroup structure of orthogonal groups of small dimension over finite fields

How much is known about the subgroup structure of the orthogonal groups (of dimension n<=7, say) over finite fields? Can anyone point me in the direction of a good reference? I'm aware of a book by ...
1
vote
1answer
122 views

Natural actions of quotients of automorphism groups

I've stumbled upon a construction which seems to be very much classical, and yet I found nothing definite about it so far in available sources. Let $\Lambda$ be a normal subgroup of the automorphism ...
4
votes
3answers
228 views

multiply transitive groups

It seems to be well-known that the six-transitive finite groups are the symmetric and alternating groups, and the only other four-transitive finite groups are the Mathieu groups (the statement can be ...
3
votes
1answer
81 views

On Groups of Maximal Class: Reference

I will be happy if one gives references (oncluding current research) for `classification' (structure) of $p$-groups of maximal class which contain abelian maximal subgroup (i.e. abelian subgroup of ...
2
votes
1answer
162 views

Every abelian torsion-free group is strictly totally orderable (via the compactness theorem)

Let $\mathbb G = (G, +)$ be a group. We say that $\mathbb G$ is strictly totally orderable (others would say bi-orderable) if there exists a total order $\preceq$ on $G$ such that $x+z \prec y + z$ ...
4
votes
3answers
412 views

Reference for Ring Structure on Group Cohomology

As a graded $\mathbb{Z}$-module, the structure of the group cohomology $H^{*}(\mathbb{Z}/n\mathbb{Z};\mathbb{Z})$ is extremely well-known. Yet, I am having difficulty finding a reference concerning ...
7
votes
4answers
411 views

What are the best settings for the large scale geometry of locally compact groups?

My current research involves locally compact groups and from time to time I am tempted to check whether certain notions and statements of geometric group theory of finitely generated groups are still ...
0
votes
1answer
135 views

Reference on elements of finite order in principal congruence subgroups of symplectic groups

We should start with the definition of the symplectic group for an arbitrary ring $R$. The symplectic group $Sp(g,R)$ is the subgroup of $SL(2g,R)$ such that all elements satisfy $M=J_g^t M J_g$ with ...
2
votes
2answers
128 views

reference request for character theory of p-extraspecial groups

In a recent preprint 1302.1929 my co-authors and I make use of some results about the character theory of extraspecial groups. These were largely gleaned from haphazard Googling, comments made by ...
12
votes
0answers
248 views

Relation Between Truncated Braid Groups and Regular Tilings of the Complex and Hyperbolic Plane

This is perhaps a vague question, but hopefully there exists literature on the subject. The question is motivated by an answer I gave to this question on math.SE. There exists a rather remarkable ...
3
votes
1answer
158 views

Groups with special automorphism group

I am looking for all finite groups $G$ such that for each subgroup $H$ of $G$ and each automorphism $\sigma$ of $H$ there exists an automorphism $\psi$ of $G$ whose restriction to $H$ is $\sigma$. Is ...
2
votes
1answer
101 views

Amenable group rings embeddable in skew fields

I've made this question on math.stackexchange.com (also offering a bounty) but I did not receive any answer: I'm looking for a reference of the following fact: given a (countable?) amenable group ...
6
votes
1answer
421 views

Growth of Thompson's group $F$

EDIT(August 2013): I accepted Mark's answer as being the state of art- there are two relevant references, one in the answer and one in the comments. The minimal growth rate of $F$ remains unknown with ...
4
votes
4answers
508 views

A catalog of faithful representations of finite groups?

I want a reference that catalogs the smallest-dimensional faithful representation of every noteworthy finite group. Specifically, I want representations on $\mathbb{R}^n$ and $\mathbb{C}^n$. Where ...