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1d

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Relations between Arboreal Group Theory and Tree Group Actions?
"Examples of such constructions are the Thompson groups": I'm afraid this is not quite true: every action of the Thompson group $F$ on a locally finite rooted tree (by rooted automorphisms) factors through the abelianization $\mathbf{Z}^2$, while every action of Thompson's group $T$ or $V$ (or any infinite simple group) is trivial (= identically trivial). 
Apr 11 
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Non split extension isomorphic (as a group) to a split extension
About an easy infinite example: Let $A$ be the product (of order 8) of cyclic groups of order 2 and 4. Let $G$ be the (restricted or unrestricted) direct product of countably many copies of $A$. Then $G$ is isomorphic to $G\times G$ and can also obviously be decomposed as a nonsplit extension of $G$ by $G$. 
Apr 11 
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Non split extension isomorphic (as a group) to a split extension
user.math.uzh.ch/ayoub/PDFFiles/DIRECT.PDF (J. Ayoub, The direct extension theorem. J. Group Theory 9 (2006), 307316.) 
Apr 10 
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Non split extension isomorphic (as a group) to a split extension
More generally than the abelian case, J. Ayoub proved that a finite group that is a direct product $A\times B$ cannot be written as a nonsplit extension of $A$ by $B$. 
Apr 10 
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Is there a generalization of niradicals in Lie algebras?
"constructing a Lie algebra like $\mathfrak{n}_\oplus\mathfrak{l}\oplus\mathfrak{n}$" makes little sense if you don't specify the brackets between elements of $\mathfrak{n}_$ and $\mathfrak{n}$. 
Apr 8 
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automorphism of finitly generated group
The BaumslagSolitar group $BS(2,4)$ has an infinitely generated automorphism group (D.J. Collins, F. Levin, Automorphisms and hopﬁcity of certain BaumslagSolitar groups, Arch. Math. 40 (1983), 385–400.) 
Apr 8 
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Characters and conjugacy classes
@QiaochuYuan: there are indeed such groups, but cardinality has nothing to do here; for instance $SL_n$ of a field of cardinal greater than continuum has a faithful finitedimensional representation. On the other hand, finitely generated simple groups have no nontrivial finitedimensional representation over any field (and over any commutative ring as well). 
Apr 7 
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Standard satisfiability for sentences in the language of ordered Abelian groups!
Maybe I misunderstand your question, but in the ordered abelian group $\mathbf{Z}$ satisfies $\exists x\forall y, y>0\Leftrightarrow y\ge x$, but the ordered abelian group $\mathbf{R}$ doesn't satisfy this sentence. Actually no $\mathbf{R}^J$ satisfies this sentence... 
Apr 5 
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Lattices in general totally disconnected locally compact groups
@Vahid: the direct product of two nondiscrete topologically simple LC groups is not simple, although it contains no open normal subgroup except itself. 
Apr 5 
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Lattices in general totally disconnected locally compact groups
The point in this paper is that the TDLC group is simple. Otherwise it's rather trivial to find TDLC groups without lattices, e.g. nonunimodular LC groups, or any LC group having $\mathbf{Q}_p^n$ as open subgroup. 
Apr 4 
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Given a set of generators of a group G, is there a method to find a presentation for G using those generators?
Then the answer is yes, it's just the standard way of changing the presentation when we change the generators, it can be found on standard textbooks such as LyndonSchupp. 
Apr 2 
awarded  Yearling 
Apr 2 
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Does there exists a finitely presented group with Dehn function > n^3 and all asymptotic cones simply connected
Yes: the free $k$step nilpotent group on $d\ge 2$ generators for $k\ge 3$, which has Dehn function $\simeq n^{k+1}$ (easy and wellknown) and simply connected asymptotic cones (Pansu). 
Apr 1 
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Homomorphisms between groups of Hermitian type and Hodge type and orthogonality
For Hermitian: are you sure that "maximal abelian" is not "maximal compact"? (otherwise $SL_2(\mathbf{R})/SO(2)$ and its powers seem to be the only examples). And is $H$ semisimple? For Hodge type: is "Admiral" a typo? 
Apr 1 
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Homomorphisms between groups of Hermitian type and Hodge type and orthogonality
Could you define "group of Hodge/Hermitian type"? the terminology is not standard and not included in basis linear algebraic group textbooks. 
Mar 30 
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Isometry groups acting transitively
It's not true that if $X$ is locally compact then so is $Isom(X)$ with compact open topology. Example: $X$ countable set with all distances $=1$. 
Mar 30 
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Amenability at infinity
"nondiscrete" is an irrelevant assumption, since given a discrete group $\Gamma$ and a fixed nondiscrete compact group $K$ such as $\mathbf{Z}_p$, the group $\Gamma$ is amenable at infinity iff $\Gamma\times K$ is amenable at infinity. 
Mar 29 
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Non existence of cyclic infinite linear algebraic groups
"infinite cyclic" sounds a bit focused and unstable... the same question with "infinite finitely generated" would be more natural. 
Mar 29 
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Non existence of cyclic infinite linear algebraic groups
@Aakumadula: could you explain why $T(k)$ can't be infinite cyclic when $T$ is an anisotropic torus? Your argument does not seem to include this case. 
Mar 28 
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Topological groups defined by completely disconnected subgroups
In general it's sort or hopeless. For instance, the topological additive group of $p$adics $\mathbf{Z}_p$ has only the trivial subgroup as a discrete subgroup. The same holds for the infinite product $G$ of infinitely many copies of $\mathbf{Z}_p$. Now consider a noncontinuous homomorphism $f:G\to C_p$, where $C_p=\mathbf{Z}/p\mathbf{Z}$, and endow $G$ with a new finer topology, namely the topology induced by the diagonal embedding into $G\times C_p$. Then the new topology also has no nontrivial discrete subgroup. Both are Hausdorff. 