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In [[mathematics]], a  [[group (mathematics)|group]] is called '''boundedly generated'''  if it can be expressed as a finite product of [[cyclic group|cyclic]] [[subgroup]]s. The property of bounded generation is also closely related with the [[congruence subgroup#Congruence subgroups and topological groups|congruence subgroup problem]] (see {{harvnb|Lubotzky|Segal|2003}}).
 
== Definitions ==
 
A group ''G'' is called ''boundedly generated'' if there exists a finite subset ''S'' of ''G'' and a positive integer ''m'' such that every element ''g'' of ''G'' can be represented as a product of at most ''m'' powers of the elements of ''S'':
: <math>g = s_1^{k_1} \cdots s_m^{k_m},</math> where <math>s_i \in S</math> and <math>k_i</math> are integers.
 
The finite set ''S'' generates ''G'', so  a boundedly generated group is [[finitely generated group|finitely generated]].
 
An equivalent definition can be given in terms of cyclic subgroups. A group ''G'' is called ''boundedly generated'' if there is a finite family ''C''<sub>1</sub>, …, ''C''<sub>''M''</sub> of not necessarily distinct [[cyclic group|cyclic]] subgroups such that ''G'' = ''C''<sub>1</sub>…''C''<sub>''M''</sub> as a set.
 
== Properties ==
 
* Bounded generation is unaffected by passing to a subgroup of [[index of a subgroup|finite index]]: if ''H'' is a finite index subgroup of ''G'' then ''G'' is boundedly generated if and only if ''H'' is boundedly generated.
* Any [[quotient group]] of a boundedly generated group is also boundedly generated.
* A [[Finitely generated group|finitely generated]] [[periodic group]] must be ''finite'' if it is boundedly generated; equivalently, an ''infinite'' finitely generated periodic group is not boundedly generated.
 
A ''pseudocharacter'' on a discrete group ''G'' is defined to be a real-valued function ''f'' on a ''G'' such that
: ''f''(''gh'') &minus; ''f''(''g'') &minus; ''f''(''h'') is uniformly bounded and ''f''(''g''<sup>''n''</sup>) = ''n''·''f''(''g'').
 
* The vector space of pseudocharacters of a boundedly generated group ''G'' is finite dimensional.
 
== Examples ==
* If ''n'' ≥ 3, the group ''SL''<sub>''n''</sub>('''Z''') is boundedly generated by its ''elementary subgroups,'' formed by matrices differing from the identity matrix only in one off-diagonal entry. In 1984, Carter and Keller gave an elementary proof of this result, motivated by a question in [[algebraic K-theory]].
* A [[free group]] on at least two generators is not boundedly generated (see below).
* The group ''SL''<sub>''2''</sub>('''Z''') is not boundedly generated, since it contains a free subgroup with two generators of index 12.
* A [[Gromov-hyperbolic group]] is boundedly generated if and only if it is ''virtually cyclic'' (or ''elementary''), i.e. contains a cyclic subgroup of finite index.
 
== Free groups are not boundedly generated ==
Several authors have stated in the mathematical literature that it is obvious that finitely generated free groups are not boundedly generated. This section contains various obvious and less obvious ways of proving this. Some of the methods, which touch on bounded cohomology, are important because they are geometric rather than algebraic, so can be applied to a wider class of groups, for example [[Gromov-hyperbolic group]]s.
 
Since for any ''n'' ≥ 2, the [[free group]] on 2 generators ''F''<sub>2</sub>  contains the free group on ''n'' generators ''F''<sub>''n''</sub> as a subgroup of finite index (in fact ''n'' – 1), once one non-cyclic free group on finitely many generators is known to be not boundedly generated, this will be true for all of them. Similarly, since ''SL''<sub>2</sub>('''Z''') contains  ''F''<sub>2</sub>  as a subgroup of index 12, it is enough to consider ''SL''<sub>2</sub>('''Z'''). In other words, to show that no ''F''<sub>''n''</sub> with ''n'' ≥ 2 has bounded generation, it is sufficient to prove this for one of them or even just for ''SL''<sub>2</sub>('''Z''') .
 
===Burnside couterexamples===
Since bounded generation is preserved under taking homomorphic images, if a single [[finitely generated group]] with at least two generators is known to be not boundedly generated, this will be true for the free group on the same number of generators, and hence for all free groups. To show that no (non-cyclic) free group has bounded generation, it is therefore enough to produce one example of a finitely generated group which is not boundedly generated, and any finitely generated infinite [[periodic group]] will work. The existence of such groups constitutes [[Golod–Shafarevich theorem|Golod and Shafarevich]]'s negative solution of the [[Burnside problem|generalized Burnside problem]] in 1964; later, other explicit examples of infinite finitely generated periodic groups were constructed by Aleshin, Olshanskii, and Grigorchuk, using [[automata theory|automata]]. Consequently, free groups of rank at least two are not boundedly generated.
 
===Symmetric groups===
The [[symmetric group]] ''S''<sub>''n''</sub> can be generated by two elements, a 2-cycle and an ''n''-cycle, so that it is a quotient group of  ''F''<sub>2</sub>. On the other hand, it is easy to show that the maximal order ''M''(''n'') of an element in ''S''<sub>''n''</sub> satisfies
 
: log ''M''(''n'') ≤ n/e
 
([[Edmund Landau]] proved the more precise asymptotic estimate log ''M''(''n'') ~ (''n'' log ''n'')<sup>1/2</sup>). In fact if the cycles in a [[cycle decomposition (group theory)|cycle decomposition]] of a [[permutation]] have length ''N''<sub>1</sub>, ..., ''N''<sub>''k''</sub> with ''N''<sub>1</sub> + ··· + ''N''<sub>''k''</sub> = ''n'', then the order of the permutation divides the product ''N''<sub>1</sub> ···''N''<sub>''k''</sub>, which in turn is bounded by (''n''/''k'')<sup>''k''</sup>, using the [[inequality of arithmetic and geometric means]]. On the other hand, (''n''/''x'')<sup>''x''</sup> is maximized when ''x''=''e''. If ''F''<sub>2</sub> could be written as a product of ''m'' cyclic subgroups, then necessarily ''n''! would have to be less than or equal to ''M''(''n'')<sup>''m''</sup> for all ''n'', contradicting [[Stirling's approximation|Stirling's asymptotic formula]].
 
===Hyperbolic geometry===
There is also a simple geometric proof that that ''G'' = ''SL''<sub>2</sub>('''Z''') is not boundedly generated. It acts by [[Möbius transformation]]s on the [[upper half-plane]] '''H''', with the [[Poincaré metric]]. Any [[compact support|compactly supported]] [[1-form]] α on a [[fundamental domain]] of ''G'' extends uniquely to a ''G''-invariant 1-form on '''H'''. If ''z'' is in '''H''' and γ is the [[geodesic]] from ''z'' to ''g''(''z''), the function defined by
:<math>  F(g)\equiv F_{\alpha,z}(g)=\int_{\gamma}\, \alpha</math>
 
satisfies the first condition for a pseudocharacter since by the [[Stokes theorem]]
:<math>  F(gh) - F(g)-F(h) = \int_{\Delta}\, d\alpha,</math>
 
where Δ is the geodesic triangle with vertices ''z'', ''g''(''z'') and ''h''<sup>−1</sup>(''z''), and geodesics triangles have area bounded by π. The homogenized function
 
:<math>f_\alpha(g) = \lim_{n\rightarrow \infty} F_{\alpha,z}(g^n)/n</math>
 
defines a pseudocharacter, depending only on α. As is well known from the theory of [[dynamical system]]s, any orbit (''g''<sup>''k''</sup>(''z'')) of a [[Möbius transformation#Hyperbolic_transforms|hyperbolic element]] ''g'' has limit set consisting of two fixed points on the extended real axis; it follows that the geodesic segment from ''z'' to ''g''(''z'') cuts through only finitely many translates of the fundamental domain. It is therefore easy to choose α so that ''f''<sub>α</sub> equals one on a given hyperbolic element and vanishes on a finite set of other hyperbolic elements with distinct fixed points. Since ''G'' therefore has an infinite-dimensional space of pseudocharacters, it cannot be boundedly generated.
 
Dynamical properties of hyperbolic elements can similarly be used to prove that any non-elementary [[Gromov-hyperbolic group]] is not boundedly generated.
 
===Brooks pseudocharacters===
Robert Brooks gave a combinatorial scheme to produce pseudocharacters of any free group ''F''<sub>''n''</sub>; this scheme was later shown to yield
an infinite-dimensional family of pseudocharacters (see {{harvnb|Grigorchuk|1994}}). [[David Epstein|Epstein]] and Fujiwara later extended these results to all non-elementary Gromov-hyperbolic groups.
 
===Gromov boundary===
This simple [[folklore]] proof uses dynamical properties of the action of hyperbolic elements on the [[Gromov boundary]] of a [[Gromov-hyperbolic group]]. For the special case of the free group ''F''<sub>''n''</sub>, the boundary (or space of ends) can be identified with the space ''X'' of [[semi-infinite]] [[word_(group_theory)#Reduced_words|reduced words]]
 
:''g''<sub>1</sub> ''g''<sub>2</sub> ···
 
in the generators and their inverses. It gives a natural compactification of the [[tree (graph theory)|tree]], given by the [[Cayley graph]] with respect to the generators. A sequence of semi-infinite words converges to another such word provided that the initial segments agree after a certain stage, so that ''X'' is compact (and [[metrizable]]). The free group acts by left multiplication on the semi-infinite words. Moreover any element ''g'' in ''F''<sub>''n''</sub> has exactly two fixed points ''g''<sup>±∞</sup>, namely the reduced infinite words given by the limits of ''g''<sup>''n''</sup> as ''n'' tends to ±∞. Furthermore ''g''<sup>''n''</sup>·''w'' tends to ''g''<sup>±∞</sub> as ''n'' tends to ±∞ for any semi-infinite word ''w''; and more generally if ''w''<sub>''n''</sub> tends to ''w''≠ ''g''<sup> ±∞</sup>, then ''g''<sup>''n''</sup>·''w''<sub>''n''</sub> tends to ''g''<sup>+∞</sub> as ''n'' tends to .  
 
If ''F''<sub>''n''</sub> were boundedly generated, it could be written as a product of cyclic groups ''C''<sub>''i''</sub>
generated by elements ''h''<sub>''i''</sub>. Let ''X''<sub>0</sub> be the countable subset given by the finitely many ''F''<sub>''n''</sub>-orbits
of the fixed points ''h''<sub>''i''</sub><sup> ±∞</sup>, the fixed points of the ''h''<sub>''i''</sub> and all their conjugates. Since ''X'' is uncountable, there
is an element of ''g'' with fixed points outside ''X''<sub>0</sub> and a point ''w'' outside ''X''<sub>0</sub> different from these fixed points. Then for
some subsequence (''g''<sub>m</sub>) of (''g''<sup>n</sup>)
 
:''g''<sub>''m''</sub> = ''h''<sub>1</sub><sup>''n''(''m'',1)</sup> ··· ''h''<sub>''k''</sub><sup>''n''(''m'',''k'')</sup>, with each ''n''(''m'',''i'') constant or strictly monotone.
 
On the one hand, by successive use of the rules for computing limits of the form ''h''<sup>''n''</sup>·''w''<sub>''n''</sub>, the limit of the right hand side applied to ''x'' is necessarily a fixed point of one of the conjugates of the ''h''<sub>''i''</sub>'s. On the other hand, this limit also must be ''g''<sup>+∞</sup>, which is not one of these points, a contradiction.
 
== References ==
*{{cite journal
|author = Carter, David and Keller, Gordon|title= Elementary expressions for unimodular matrices|journal= Comm. Alg.|volume=12|year=1984|pages=379–389|doi= 10.1080/00927878408823008
|ref = harv
|issue = 4}}
 
*{{cite journal
|author= Epstein, David and Fujiwara, Koji| title = The second bounded cohomology of word-hyperbolic groups| journal=Topology|volume=36|year=1997|pages=1275–1289| doi = 10.1016/S0040-9383(96)00046-8
|ref= harv
|issue= 6}}
 
*{{cite journal | author= Ghys, Etienne and Barge, Jean| title= Surfaces et cohomologie bornée| journal=Inventiones Mathematicae| year=1988|volume=92|
pages=509–526| doi= 10.1007/BF01393745 | ref= harv | issue= 3}}
 
*{{cite journal|
author=Grigorchuk, R.I.|  title=On Burnside's problem on periodic groups|journal= Functional Anal. Appl.|volume= 14|year =1980|pages=41–43|
ref=harv}}
 
*{{cite journal
|author= Grigorchuk, R.I.|title=Some results in bounded cohomology |journal =London Mathematical Society Lecture Note Series|volume=224|year =1994|pages=111–163
|ref= harv
|isbn= 0-521-46595-8}}
 
*{{cite book
|author = Landau, Edmund |title =Handbuch der Lehrer von der Verteilung der Primzahlen, Vol. I|publisher =Chelsea|year= 1974| isbn =  0-8284-0096-2}} (see pages 222-229, also available on the [[arXiv|Cornell archive]])
 
*{{Cite journal
|last = [[Alexander Lubotzky|Lubotzky]]|first=Alexander|last2=Segal|first2= Dan | title = Subgroup growth | series= Progress in Mathematics| publisher=Birkhäuser|year=2003
|ref = harv
|postscript = <!--None-->
|isbn = 3-7643-6989-2}}.
 
*{{cite journal | author = Polterovich, Leonid and Rudnick, Zeev| title= Stable mixing for cat maps and quasi-morphisms of the modular group| year=2004| journal = Erg. Th. & Dynam. Syst.| volume=24|pages=609–619| doi= 10.1017/S0143385703000531 | ref = harv | issue = 2}}
 
[[Category:Algebra]]
[[Category:Geometric group theory]]

Latest revision as of 20:43, 4 October 2012

In mathematics, a group is called boundedly generated if it can be expressed as a finite product of cyclic subgroups. The property of bounded generation is also closely related with the congruence subgroup problem (see Template:Harvnb).

Definitions

A group G is called boundedly generated if there exists a finite subset S of G and a positive integer m such that every element g of G can be represented as a product of at most m powers of the elements of S:

g=s1k1smkm, where siS and ki are integers.

The finite set S generates G, so a boundedly generated group is finitely generated.

An equivalent definition can be given in terms of cyclic subgroups. A group G is called boundedly generated if there is a finite family C1, …, CM of not necessarily distinct cyclic subgroups such that G = C1CM as a set.

Properties

  • Bounded generation is unaffected by passing to a subgroup of finite index: if H is a finite index subgroup of G then G is boundedly generated if and only if H is boundedly generated.
  • Any quotient group of a boundedly generated group is also boundedly generated.
  • A finitely generated periodic group must be finite if it is boundedly generated; equivalently, an infinite finitely generated periodic group is not boundedly generated.

A pseudocharacter on a discrete group G is defined to be a real-valued function f on a G such that

f(gh) − f(g) − f(h) is uniformly bounded and f(gn) = n·f(g).
  • The vector space of pseudocharacters of a boundedly generated group G is finite dimensional.

Examples

  • If n ≥ 3, the group SLn(Z) is boundedly generated by its elementary subgroups, formed by matrices differing from the identity matrix only in one off-diagonal entry. In 1984, Carter and Keller gave an elementary proof of this result, motivated by a question in algebraic K-theory.
  • A free group on at least two generators is not boundedly generated (see below).
  • The group SL2(Z) is not boundedly generated, since it contains a free subgroup with two generators of index 12.
  • A Gromov-hyperbolic group is boundedly generated if and only if it is virtually cyclic (or elementary), i.e. contains a cyclic subgroup of finite index.

Free groups are not boundedly generated

Several authors have stated in the mathematical literature that it is obvious that finitely generated free groups are not boundedly generated. This section contains various obvious and less obvious ways of proving this. Some of the methods, which touch on bounded cohomology, are important because they are geometric rather than algebraic, so can be applied to a wider class of groups, for example Gromov-hyperbolic groups.

Since for any n ≥ 2, the free group on 2 generators F2 contains the free group on n generators Fn as a subgroup of finite index (in fact n – 1), once one non-cyclic free group on finitely many generators is known to be not boundedly generated, this will be true for all of them. Similarly, since SL2(Z) contains F2 as a subgroup of index 12, it is enough to consider SL2(Z). In other words, to show that no Fn with n ≥ 2 has bounded generation, it is sufficient to prove this for one of them or even just for SL2(Z) .

Burnside couterexamples

Since bounded generation is preserved under taking homomorphic images, if a single finitely generated group with at least two generators is known to be not boundedly generated, this will be true for the free group on the same number of generators, and hence for all free groups. To show that no (non-cyclic) free group has bounded generation, it is therefore enough to produce one example of a finitely generated group which is not boundedly generated, and any finitely generated infinite periodic group will work. The existence of such groups constitutes Golod and Shafarevich's negative solution of the generalized Burnside problem in 1964; later, other explicit examples of infinite finitely generated periodic groups were constructed by Aleshin, Olshanskii, and Grigorchuk, using automata. Consequently, free groups of rank at least two are not boundedly generated.

Symmetric groups

The symmetric group Sn can be generated by two elements, a 2-cycle and an n-cycle, so that it is a quotient group of F2. On the other hand, it is easy to show that the maximal order M(n) of an element in Sn satisfies

log M(n) ≤ n/e

(Edmund Landau proved the more precise asymptotic estimate log M(n) ~ (n log n)1/2). In fact if the cycles in a cycle decomposition of a permutation have length N1, ..., Nk with N1 + ··· + Nk = n, then the order of the permutation divides the product N1 ···Nk, which in turn is bounded by (n/k)k, using the inequality of arithmetic and geometric means. On the other hand, (n/x)x is maximized when x=e. If F2 could be written as a product of m cyclic subgroups, then necessarily n! would have to be less than or equal to M(n)m for all n, contradicting Stirling's asymptotic formula.

Hyperbolic geometry

There is also a simple geometric proof that that G = SL2(Z) is not boundedly generated. It acts by Möbius transformations on the upper half-plane H, with the Poincaré metric. Any compactly supported 1-form α on a fundamental domain of G extends uniquely to a G-invariant 1-form on H. If z is in H and γ is the geodesic from z to g(z), the function defined by

F(g)Fα,z(g)=γα

satisfies the first condition for a pseudocharacter since by the Stokes theorem

F(gh)F(g)F(h)=Δdα,

where Δ is the geodesic triangle with vertices z, g(z) and h−1(z), and geodesics triangles have area bounded by π. The homogenized function

fα(g)=limnFα,z(gn)/n

defines a pseudocharacter, depending only on α. As is well known from the theory of dynamical systems, any orbit (gk(z)) of a hyperbolic element g has limit set consisting of two fixed points on the extended real axis; it follows that the geodesic segment from z to g(z) cuts through only finitely many translates of the fundamental domain. It is therefore easy to choose α so that fα equals one on a given hyperbolic element and vanishes on a finite set of other hyperbolic elements with distinct fixed points. Since G therefore has an infinite-dimensional space of pseudocharacters, it cannot be boundedly generated.

Dynamical properties of hyperbolic elements can similarly be used to prove that any non-elementary Gromov-hyperbolic group is not boundedly generated.

Brooks pseudocharacters

Robert Brooks gave a combinatorial scheme to produce pseudocharacters of any free group Fn; this scheme was later shown to yield an infinite-dimensional family of pseudocharacters (see Template:Harvnb). Epstein and Fujiwara later extended these results to all non-elementary Gromov-hyperbolic groups.

Gromov boundary

This simple folklore proof uses dynamical properties of the action of hyperbolic elements on the Gromov boundary of a Gromov-hyperbolic group. For the special case of the free group Fn, the boundary (or space of ends) can be identified with the space X of semi-infinite reduced words

g1 g2 ···

in the generators and their inverses. It gives a natural compactification of the tree, given by the Cayley graph with respect to the generators. A sequence of semi-infinite words converges to another such word provided that the initial segments agree after a certain stage, so that X is compact (and metrizable). The free group acts by left multiplication on the semi-infinite words. Moreover any element g in Fn has exactly two fixed points g±∞, namely the reduced infinite words given by the limits of gn as n tends to ±∞. Furthermore gn·w tends to g±∞ as n tends to ±∞ for any semi-infinite word w; and more generally if wn tends to wg ±∞, then gn·wn tends to g+∞ as n tends to ∞.

If Fn were boundedly generated, it could be written as a product of cyclic groups Ci generated by elements hi. Let X0 be the countable subset given by the finitely many Fn-orbits of the fixed points hi ±∞, the fixed points of the hi and all their conjugates. Since X is uncountable, there is an element of g with fixed points outside X0 and a point w outside X0 different from these fixed points. Then for some subsequence (gm) of (gn)

gm = h1n(m,1) ··· hkn(m,k), with each n(m,i) constant or strictly monotone.

On the one hand, by successive use of the rules for computing limits of the form hn·wn, the limit of the right hand side applied to x is necessarily a fixed point of one of the conjugates of the hi's. On the other hand, this limit also must be g+∞, which is not one of these points, a contradiction.

References

  • One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

    In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

    Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

    Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

    A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

    The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

    There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
  • One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

    In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

    Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

    Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

    A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

    The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

    There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
  • One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

    In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

    Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

    Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

    A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

    The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

    There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
  • One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

    In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

    Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

    Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

    A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

    The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

    There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
  • One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

    In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

    Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

    Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

    A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

    The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

    There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
  • One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

    In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

    Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

    Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

    A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

    The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

    There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang.
  • One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

    In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

    Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

    Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

    A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

    The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

    There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang