* Kurds say their advance has gained regional support

* Kurdish militia deny Iran, Iraq, Assad helping them

* Rebels now squeezed on both sides in northern Syria

* Kurdish gains could provoke wider regional tensions

By Erika Solomon and Isabel Coles

BEIRUT/ARBIL, Nov 11 (Reuters) - With a string of military gains across northeastern Syria, a Kurdish militia is solidifying a geographic and political presence in the war-torn country, posing a dilemma for regional powers.

Long oppressed under Syrian President Bashar al-Assad and his father before him, Kurds view the civil war as an opportunity to gain the kind of autonomy enjoyed by their ethnic kin in neighbouring Iraq.

But their offensive has stirred mixed feelings, globally, regionally and locally, even among some fellow Kurds, who say the Kurdish fighters have drifted into a regional axis supporting Assad, something they deny.

To Assad and his Shi'ite allies, their gains mean more territory out of Sunni rebel hands two and a half years into a revolt against his rule.

Foreign powers supporting the opposition, meanwhile, hope they will deliver a blow to al Qaeda-linked fighters, whose rising power in northern Syria had gone unchecked for months.

"The advance has basically been accepted by all," said Piroz Perik, a Kurdish activist from the town of Qamishli.

Such statements overlook widespread concerns over the impact of the Kurdish militia gains in a conflict that not only threatens Syria's unity, but the stability of neighbouring countries with similar ethnic and sectarian divisions.

Numbering more than 25 million, non-Arab Kurds are often described as the world's largest ethnic group without a state. Territories where they predominate, which they call Kurdistan, span parts of Turkey and Iran as well as Syria and Iraq.

Turkey began digging foundations for a wall along part of its border with Syria last month, citing security reasons but prompting protests from Kurds who said it was aimed at preventing closer cross-border ties between their communities.

Rising Kurdish assertiveness in Syria puts Turkey in a particularly tough position as it tries to make peace on its own soil with the Kurdistan Workers Party (PKK), which fought for greater Kurdish autonomy in Turkey for three decades.

The power grab by the Syrian Kurdish militia associated with the Democratic Union Party, or PYD, could embolden the PKK, with which it is closely aligned. But Turkey is also uneasy with having al Qaeda-linked groups on its doorstep.

"What you are going to see is a clearer division of northern Syria between the PYD and Islamist rebel forces," said Kurdish activist Perik.

Redur Xelil, a spokesman for Kurdish militias linked to the PYD, said more than two-thirds of Syrian Kurdish territory had been captured, mostly in northern Hassaka province, where Kurds make up 70 percent of the population and Arabs the rest.

Xelil also hinted the militia could try to take northern towns where Kurds are a minority compared to Arabs, such as the strategic border towns of Jarablus and Azaz that rebels have used as supply routes from Turkey.

Such a move would likely prompt a fierce response from the mostly-Sunni Arab rebels.

"I'm not saying we will do it. Let's take things as they come. We are waiting to see if the armed groups (rebels) will ensure safe movement for Kurds in that region first," he said.

ISLAMIST RETREAT

Tel Abyad, now in the Kurds' sights, is an important cross-border route with Turkey used for supplies. Ras al-Ain, the frontier town already taken by the Kurds this week, played a similar role.

The advances call into question the relative strength of the rebels - particularly the al Qaeda-linked Nusra Front and the Islamic State of Iraq and the Levant (ISIL), which have been imposing their will across rebel-held territory.

The two groups also led offensives on Kurdish areas that meant control swung between hardline Islamist rebels and the Kurds for months.

But Islamist rebel sources said it was unlikely that they would seek to challenge Kurdish control now, saying that the balance of power in the northeast has been set and they would focus on the northwest for now.

"The Nusra Front and the Islamic State of Iraq and the Levant (ISIL) had to leave because of important fronts to defend in Aleppo," said a source with ties to hardline Islamist units.

Rebels also argue that their defeat says more about who was helping their foes than the strength of the Kurdish forces themselves. They said help from Assad's forces and Shi'ite-led Iraq was the reason for Kurdish gains.

Such statements echo local accounts of the PYD's capture of Yaaroubiya on the eastern border with Iraq, where fighters began their push on Kurdish territory last month.

Residents there told Reuters the battle had gone on for four days until Iraqi forces joined the fight, using troops and artillery to turn the tide.

The Iraqi government strongly denies supporting any faction in Syria, including Kurds.

REGIONAL ASSAD AXIS

Although Kurds in Syria say they are against Assad and do not seek a separate state, they are wary of the Arab-led revolt.

The PYD is seen as more pragmatic and open to cooperating with any group to reach goals of autonomy and increasing its own power.

Its opponents, both Kurdish and Arab, say the recent gains clearly show the PYD had drifted into a regional Shi'ite axis behind Assad, whose Alawite sect is a Shi'ite offshoot.

A Kurdish political source in Syria said that the PYD offensive was timed to coincide with a push by Assad's forces to the northwest, near the city of Aleppo.

"Assad's forces also organised Arab militias in the area, most of them tribesmen at odds with al Qaeda's growing power here," he said. "They fought alongside the Kurds."

Xelil denied his fighters worked with outside groups.

But a senior Iraqi politician said Shi'ite power Iran, Assad's main regional ally, was also actively backing the PYD.

"Iran supports these groups to guarantee having a powerful group in Syria in case things go out of control," he said, adding that Tehran was creating a network of allies from minority groups across the country to bolster their interests and to create alternative partners should Assad fall.

The Iraqi politician said Baghdad's Shi'ite government was supporting the Kurds to weaken cross-border ties among Sunnis.

"(They) may help them in cooperation with Iran to create an autonomous Kurdish region ... to establish a buffer zone between Iraqi and Syrian Sunnis."

EYE ON GENEVA

For the PYD itself, the strategic aims of its battlefield gains may be political as much as military. It is trying to secure its role as the dominant power among Syria's Kurds.

Some Kurdish sources say the PYD is hoping its gains will make a case for it to have a spot at the table at planned Syria talks, originally set to be held in November.

The disparate Western-backed Syrian opposition has included some Kurdish members that oppose the PYD, which they accuse of seeking to replace Assad's one-party rule with its own and imposing itself by force. Many Kurds on the ground, however, see the PYD as protectors.

Russia and the United States have been working to get Syria's warring parties to "Geneva 2" in order to create a transitional government and put an end to fighting that has killed well over 100,000 people.

Intransigence on both sides and differing views from Moscow, Assad's longtime arms supplier, and Washington, which backs the opposition, has delayed a date for talks.

But foreign officials say they still hope to hold negotiations by the end of 2013.

"It is our right to self-administration in the Kurdish areas. We're not asking for separation, simply the right to manage our affairs," said PYD spokesman Xelil.

"If Geneva 2 is to organize the future Syria, then we have a right to our own representation."

(Additional reporting by Suadad al-Salhy in Baghdad, Ziad al-Sinjary in Mosul, Nick Tattersall and Dasha Afanasieva in Istanbul and Khaled Yacoub Oweis in Amman; editing by Philippa Fletcher)

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