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In [[mathematics]], a '''totally disconnected group''' is a [[topological group]] that is [[totally disconnected]]. Such topological groups are necessarily [[Hausdorff space|Hausdorff]]. | |||
Interest centres on [[locally compact]] totally disconnected groups (variously referred to as groups of '''td-type''',<ref>{{harvnb|Cartier|1979|loc=§1.1}}</ref> [[locally profinite group]]s,<ref name=BushnellHenniart>{{harvnb|Bushnell|Henniart|2006|loc=§1.1}}</ref> '''t.d. groups'''<ref>{{harvnb|Borel|Wallach|2000|loc=Chapter X}}</ref>). The [[compact space|compact]] case has been heavily studied – these are the [[profinite group]]s – but for a long time not much was known about the general case. A theorem of [[David van Dantzig|van Dantzig]] from the 1930s, stating that every such group contains a compact [[open set|open]] [[subgroup]], was all that was known. Then groundbreaking work on this subject was done in 1994, when [[George Willis (mathematician)|George Willis]] showed that every locally compact totally disconnected group contains a so-called ''tidy'' subgroup and a special function on its automorphisms, the ''scale function'', thereby advancing the knowledge of the local structure. Advances on the ''global structure'' of totally disconnected groups have been obtained in 2011 by Caprace and Monod, with notably a clasification of [[characteristically simple group]]s and of Noetherian groups. | |||
==Locally compact case== | |||
{{main|Locally profinite group}} | |||
In a locally compact, totally disconnected group, every [[neighbourhood (mathematics)|neighbourhood]] of the identity contains a compact open subgroup. Conversely, if a group is such that the identity has a [[neighbourhood basis]] consisting of compact open subgroups, then it is locally compact and totally disconnected.<ref name=BushnellHenniart/> | |||
===Tidy subgroups=== | |||
Let ''G'' be a locally compact, totally disconnected group, ''U'' a compact open subgroup of ''G'' and <math>\alpha</math> a continuous automorphism of ''G''. | |||
Define: | |||
:<math>U_{+}=\bigcap_{n\ge 0}\alpha^n(U)</math> | |||
:<math>U_{-}=\bigcap_{n\ge 0}\alpha^{-n}(U)</math> | |||
:<math>U_{++}=\bigcup_{n\ge 0}\alpha^n(U_{+})</math> | |||
:<math>U_{--}=\bigcup_{n\ge 0}\alpha^{-n}(U_{-})</math> | |||
''U'' is said to be '''tidy''' for <math>\alpha</math> if and only if <math>U=U_{+}U_{-}=U_{-}U_{+}</math> and <math>U_{++}</math> and <math>U_{--}</math> are closed. | |||
===The scale function=== | |||
The index of <math>\alpha(U_{+})</math> in <math>U_{+}</math> is shown to be finite and independent of the ''U'' which is tidy for <math>\alpha</math>. Define the scale function <math>s(\alpha)</math> as this index. Restriction to [[inner automorphism]]s gives a function on ''G'' with interesting properties. These are in particular:<br> | |||
Define the function <math>s</math> on ''G'' by | |||
<math>s(x):=s(\alpha_{x})</math>, | |||
where <math>\alpha_{x}</math> is the inner automorphism of <math>x</math> on ''G''. | |||
<math>s</math> is continuous.<br> | |||
<math>s(x)=1</math>, whenever x in ''G'' is a compact element.<br> | |||
<math>s(x^n)=s(x)^n</math> for every integer <math>n</math>.<br> | |||
The modular function on ''G'' is given by <math>\Delta(x)=s(x)s(x^{-1})^{-1}</math>.<br> | |||
===Calculations and applications=== | |||
The scale function was used to prove a conjecture by Hofmann and Mukherja and has been explicitly calculated for [[p-adic]] [[Lie group]]s and linear groups over local skew fields by Helge Glöckner. | |||
==Notes== | |||
{{reflist}} | |||
==References== | |||
*{{Citation | |||
| last=Borel | |||
| first=Armand | |||
| author-link=Armand Borel | |||
| last2=Wallach | |||
| first2=Nolan | |||
| author2-link=Nolan Wallach | |||
| title=Continuous cohomology, discrete subgroups, and representations of reductive groups | |||
| year=2000 | |||
| edition=Second | |||
| publisher=[[American Mathematical Society]] | |||
| location=Providence, Rhode Island | |||
| series=Mathematical surveys and monographs | |||
| volume=67 | |||
| isbn=978-0-8218-0851-1 | |||
| mr=1721403 | |||
}} | |||
*{{Citation | last1=Bushnell | first1=Colin J. | last2=Henniart | first2=Guy | title=The local Langlands conjecture for GL(2) | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] | isbn=978-3-540-31486-8 | doi=10.1007/3-540-31511-X | mr=2234120 | year=2006 | volume=335}} | |||
*{{Citation | last1=Caprace | first1=Pierre-Emmanuel | last2=Monod | first2=Nicolas | title=Decomposing locally compact groups into simple pieces | journal=Math. Proc. Cambridge Philos. Soc. | doi=10.1017/S0305004110000368 | mr=2739075 | year=2011 | volume=150 | pages=97–128}} | |||
*{{Citation | |||
| last=Cartier | |||
| first=Pierre | |||
| author-link=Pierre Cartier (mathematician) | |||
| contribution=Representations of <math>\mathfrak{p}</math>-adic groups: a survey | |||
| year=1979 | |||
| title=Automorphic Forms, Representations, and L-Functions | |||
| editor1-last=Borel | |||
| editor1-first=Armand | |||
| editor1-link=Armand Borel | |||
| editor2-last=Casselman | |||
| editor2-first=William | |||
| editor2-link=William Casselman (mathematician) | |||
| url=http://www.ams.org/online_bks/pspum331/pspum331-ptI-7.pdf | |||
| publisher=[[American Mathematical Society]] | |||
| publication-place=Providence, Rhode Island | |||
| series=Proceedings of Symposia in Pure Mathematics | |||
| volume=33, Part 1 | |||
| pages=111–155 | |||
| isbn=978-0-8218-1435-2 | |||
| mr=0546593 | |||
}} | |||
*G.A. Willis - [http://gdz.sub.uni-goettingen.de/no_cache/dms/load/img/?IDDOC=167209 The structure of totally disconnected, locally compact groups], [[Mathematische Annalen]] 300, 341-363 (1994) | |||
[[Category:Topological groups]] | |||
Revision as of 18:23, 11 January 2013
In mathematics, a totally disconnected group is a topological group that is totally disconnected. Such topological groups are necessarily Hausdorff.
Interest centres on locally compact totally disconnected groups (variously referred to as groups of td-type,[1] locally profinite groups,[2] t.d. groups[3]). The compact case has been heavily studied – these are the profinite groups – but for a long time not much was known about the general case. A theorem of van Dantzig from the 1930s, stating that every such group contains a compact open subgroup, was all that was known. Then groundbreaking work on this subject was done in 1994, when George Willis showed that every locally compact totally disconnected group contains a so-called tidy subgroup and a special function on its automorphisms, the scale function, thereby advancing the knowledge of the local structure. Advances on the global structure of totally disconnected groups have been obtained in 2011 by Caprace and Monod, with notably a clasification of characteristically simple groups and of Noetherian groups.
Locally compact case
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In a locally compact, totally disconnected group, every neighbourhood of the identity contains a compact open subgroup. Conversely, if a group is such that the identity has a neighbourhood basis consisting of compact open subgroups, then it is locally compact and totally disconnected.[2]
Tidy subgroups
Let G be a locally compact, totally disconnected group, U a compact open subgroup of G and a continuous automorphism of G.
Define:
U is said to be tidy for if and only if and and are closed.
The scale function
The index of in is shown to be finite and independent of the U which is tidy for . Define the scale function as this index. Restriction to inner automorphisms gives a function on G with interesting properties. These are in particular:
Define the function on G by
,
where is the inner automorphism of on G.
is continuous.
, whenever x in G is a compact element.
for every integer .
The modular function on G is given by .
Calculations and applications
The scale function was used to prove a conjecture by Hofmann and Mukherja and has been explicitly calculated for p-adic Lie groups and linear groups over local skew fields by Helge Glöckner.
Notes
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References
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To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010 - Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010 - Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010 - Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010 - G.A. Willis - The structure of totally disconnected, locally compact groups, Mathematische Annalen 300, 341-363 (1994)